Go to:
Settings > Apps > Select Gmail > Force stop > Disable
Android since v1.0. Linux user since 2001.
It is just a matter of how many folders you sync and if K9 kills the connections. The K9 mail app is quite good, so I imagine it is doing this correctly. Having said that, I am biased, have you tried MailDroid?
That's a little strange. Using AquaMail I sync per account up to 1000 emails to over a dozen folders without any problems.
I agree with maildroiddev that K-9 is a fine app, but I gotta think it's screwing up in this case - unless you're have a desktop email client open at the same time, such as Thunderbird or Outlook. I have seen the too many connections error in such a situation.
Android since v1.0. Linux since 2001.