Where or what is your ISP doing with those messages then? AS you say they are sending without error. Perhaps you ISP tech support can check their logs and determine exactly what they are doing with your outgoing mail after they receive it.

What ISP is it?

the problem is that the messages do mot reach the server at all, but there is clearly a link between because messages are received

bezeqint is the ISP

Please add the troubleshooting information to your post To find the Troubleshooting information:

• Open Help (or click on three-line-icon and select Help)
• Choose Troubleshooting Information
• Use the button Copy to clipboard to select all. Do not check box "Include account names"!
• Paste this in your post.

Please also add the result of these troubleshooting steps to your post

• Does Thunderbird work in TB Safe mode (see Thunderbird Safe Mode)?
• Do you use anti-virus and firewall software? What is the version?

Application Basics

   Name: Thunderbird
   Version: 52.9.1
   User Agent: Mozilla/5.0 (Windows NT 10.0; WOW64; rv:52.0) Gecko/20100101 Thunderbird/52.9.1
   Profile Folder: Open Folder

             (Local drive)
   Application Build ID: 20180709124824
   Enabled Plugins: about:plugins
   Build Configuration: about:buildconfig
   Memory Use: about:memory
   Profiles: about:profiles

 Mail and News Accounts
   account1:
     INCOMING: account1, , (pop3) mail.bezeqint.net:110, plain, passwordCleartext
     OUTGOING: , out.bezeqint.net:25, plain, passwordCleartext, true

   account2:
     INCOMING: account2, , (none) Local Folders, plain, passwordCleartext

 Crash Reports
   https://crash-stats.mozilla.com/report/index/bp-c02f4768-f5fa-474a-9981-623010180529 (5/29/2018)
   https://crash-stats.mozilla.com/report/index/bp-cddceedc-1b77-4f07-911e-9b6c60180529 (5/29/2018)
   https://crash-stats.mozilla.com/report/index/bp-1941b996-c2e9-42d8-96e3-87bce0180528 (5/28/2018)
   https://crash-stats.mozilla.com/report/index/bp-c24d32e1-abf8-4e57-adc4-756f50180528 (5/28/2018)
   https://crash-stats.mozilla.com/report/index/bp-b7f6e477-69d2-47a4-beec-b59590180528 (5/28/2018)

 Extensions
   BiDi Mail UI, 0.9.9b3, true, {54e72d7b-ed31-4736-854f-f517358f21ee}
   Lightning, 5.4.9.1, true, {e2fda1a4-762b-4020-b5ad-a41df1933103}
   WiseStamp, 3.13.44, true, wisestamp@wisestamp.com

 Important Modified Preferences

   Name: Value

     browser.cache.disk.capacity: 358400
     browser.cache.disk.filesystem_reported: 1
     browser.display.use_document_fonts: 0
     extensions.lastAppVersion: 52.9.1
     font.internaluseonly.changed: true
     font.minimum-size.x-western: 20
     font.name.monospace.el: Consolas
     font.name.monospace.x-cyrillic: Consolas
     font.name.monospace.x-unicode: Consolas
     font.name.monospace.x-western: Times New Roman
     font.name.sans-serif.el: Calibri
     font.name.sans-serif.x-cyrillic: Calibri
     font.name.sans-serif.x-unicode: Calibri
     font.name.sans-serif.x-western: Times New Roman
     font.name.serif.el: Cambria
     font.name.serif.x-cyrillic: Cambria
     font.name.serif.x-unicode: Cambria
     font.size.fixed.el: 14
     font.size.fixed.x-cyrillic: 14
     font.size.fixed.x-unicode: 14
     font.size.fixed.x-western: 20
     font.size.variable.el: 17
     font.size.variable.x-cyrillic: 17
     font.size.variable.x-unicode: 17
     font.size.variable.x-western: 20
     mail.openMessageBehavior.version: 1
     mail.winsearch.firstRunDone: true
     mailnews.database.global.datastore.id: 169b00a2-c926-422c-a3e8-5d326483653
     mailnews.database.global.views.conversation.columns: {"threadCol":{"visible":true,"ordinal":"1"},"flaggedCol":{"visible":true,"ordinal":"3"},"attachmentCol":{"visible":false…
     mailnews.database.global.views.global.columns: {"threadCol":{"visible":true,"ordinal":"3"},"flaggedCol":{"visible":true,"ordinal":"5"},"attachmentCol":{"visible":false…
     media.gmp.storage.version.observed: 1
     network.cookie.prefsMigrated: true
     network.predictor.cleaned-up: true
     places.database.lastMaintenance: 1536686334
     places.history.expiration.transient_current_max_pages: 122334
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_bgcolor: false
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_bgimages: false
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_duplex: -437918235
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_edge_bottom: 0
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_edge_left: 0
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_edge_right: 0
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_edge_top: 0
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_evenpages: true
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_footercenter:
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_footerleft: &PT
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_footerright: &D
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_headercenter:
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_headerleft: &T
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_headerright: &U
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_in_color: true
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_margin_bottom: 0.5
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_margin_left: 0.5
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_margin_right: 0.5
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_margin_top: 0.5
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_oddpages: true
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_orientation: 0
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_page_delay: 50
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_paper_data: 9
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_paper_height: -1.00
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_paper_name:
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_paper_size_unit: 1
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_paper_width: -1.00
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_resolution: 600
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_reversed: false
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_scaling: 1.00
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_shrink_to_fit: true
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_to_file: false
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_unwriteable_margin_bottom: 0
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_unwriteable_margin_left: 0
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_unwriteable_margin_right: 0
     print.printer_Xerox_WorkCentre_3210_PCL_6.print_unwriteable_margin_top: 0
     security.sandbox.content.tempDirSuffix: {5e8d9bad-6c48-400e-b232-1f7de1366408}

 Graphics

     GPU #1
     Description: Intel(R) HD Graphics 620
     Vendor ID: 0x8086
     Device ID: 0x5916
     RAM: Unknown
     Drivers: igdumdim64 igd10iumd64 igd10iumd64 igd12umd64 igdumdim32 igd10iumd32 igd10iumd32 igd12umd32
     Driver Version: 21.20.16.4550
     Driver Date: 11-11-2016

     GPU #2
     Description: AMD Radeon (TM) R7 M460
     Vendor ID: 0x1002
     Device ID: 0x6900
     RAM: 2048
     Drivers: aticfx64 aticfx64 aticfx64 amdxc64 aticfx32 aticfx32 aticfx32 amdxc32 atiumd64 atidxx64 atidxx64 atiumdag atidxx32 atidxx32 atiumdva atiumd6a atitmm64
     Driver Version: 21.19.128.4
     Driver Date: 9-3-2016

     Features
     Direct2D: false
     DirectWrite: true (10.0.17134.1)
     WebGL Renderer: Google Inc. -- ANGLE (Intel(R) HD Graphics 620 Direct3D9Ex vs_3_0 ps_3_0) -- OpenGL ES 2.0 (ANGLE 2.1.0.2a250c8a0e15)

     AzureCanvasBackend: skia
     AzureCanvasAccelerated: 0
     AzureFallbackCanvasBackend: cairo
     AzureContentBackend: skia

 JavaScript

 Incremental GC: 1

 Accessibility

   Activated: 0
   Prevent Accessibility: 0

 Library Versions

     Expected minimum version
     Version in use

     NSPR
     4.13.1
     4.13.1

     NSS
     3.28.6
     3.28.6

     NSS Util
     3.28.6
     3.28.6

     NSS SSL
     3.28.6
     3.28.6

     NSS S/MIME
     3.28.6
     3.28.6

I restarted in safe mode and it did not help. We also disabled firewall. Very frustrating!

restarted the router - it was a fluke of configs. working perfectly now. Thank you for your help!