I tried a few different ways to fix Thunderbird’s Maximum IMAP Connections exceeded issue. The most common is to go into advanced settings and change the value from 5 to a lesser value.
I only have one IMAP account, so changing to three or one didn’t fix the issue. Final resolution? Deleting the account and remaking it.
Not really a solution, but time-wise, it was much faster than to continue researching how to fix this issue in my particular case.