The nature of the problem (the intermittency, in particular) is an indication that this could be caused by you using a VPN. Try turning it off if that's the case.

Read the following article to see other possible causes of the problem:

https://support.mozilla.org/kb/cannot-send-messages

Hi, Unfortunately, none of these are relevant to this problem which has been going on for a long time. There was a period when all was well. Then began to suffer from this unpredictable intermittent failure of sending emails which is very frustrating. Have explored and checked everything possible.

However, when we can't send an email through Thunderbird if we go to the far end of our home and hook into our neighbours wifi the email will go. You might think that this points a finger at something in our house but we have a new Modem/Router device which is supposedly working properly. Any ideas?

This clearly points to a networking issue. I mentioned the possibility that it could be a VPN, but if you check that article, you'll see there are other things to look at (e.g. firewall settings). One thing you may do to further narrow the issue is connect the computer to the router with an Ethernet cable, to isolate whether the problem is in the WiFi or some other router misconfiguration.

Hi, Just tried the direct connection to the router and it did not work. Then immediately tried to send an email wirelessy which also failed. We do not have VPN. Regarding firewall settings, what should we be looking for? We are not technically minded...