See:

• http://kb.mozillazine.org/Websites_report_cookies_are_disabled

(Open the "Exceptions" list and remove any entries for the site, if found, with the status "Block".) Does this mean I have to add all the websites I use regularly and mark them allow? If so, I don't see much use for the basic privacy settings regarding cookies & 3rd party cookies being set to "allow."?

You only need to allow cookies if you let them expire if Firefox closes and you want to keep some for the next session.
If you keep cookies until they expire then you shouldn't need to set an allow exception.

Try to clear the cookies from sites that cause problems.

• Tools > Options > Privacy > Firefox will: "Use custom settings for history" > Cookies: "Show Cookies"

If clearing the cookies doesn't help then it is possible that the file cookies.sqlite that stores the cookies is corrupted.
Rename (or delete) cookies.sqlite (cookies.sqlite.old) and delete cookies.sqlite-journal and cookies.txt, if they exist, in the Firefox Profile Folder in case the file cookies.sqlite got corrupted.

• http://kb.mozillazine.org/Profile_folder_-_Firefox

Think with your help I may have found the culprit. Under Options>Privacy>History> Settings>Clear History when FireFox closes> Settings I had cookies box checked for "when I quit FireFox it should automatically clear all:" The program I had trouble with was NOT listed, so I will try it again soon. Thanks for walking me through this. Ken

Are there any websites listed in the Cookie Exception window?

• Tools > Options > Privacy > Cookies: "Exceptions"

If there are then remove them all unless you added them yourself for a specific reason.
There is usually no need to block cookies apart from disabling third-party cookies.