No such limit. What exactly happens or is not happening when adding the account, that is different from how the first five accounts were setup? Are you getting any errors? Which version of Thunderbird are you using?

If you're adding multiple accounts on the same server, first make sure each existing and new account has a distinct Account Name in Tools/Account Settings (select the account in the left pane), instead of the default, e.g. set Gmail1, Gmail2 etc. as the Account Names.

Same for the outgoing servers: give each one a distinct Description to make it easier to match the incoming account with the correct smtp.