This is probably something about your cell provider.
To get help, you will need to explain all the details:
k-9 version
type: imap-ssl, pop, and so on. and port
provider
cell carrier
when you push the sync button, what happens?
which of "we" has differences above (if there are 4 of you and it's
across 3 carriers, that's relevant)
Then, you probably should use adb logcat to see what k-9 is saying. I
realize that may be hard but I think it is what will explain things.
It might be nice for k-9 to have the equivalent of mac mail's
"connection doctor". Basically try to make outgoing and incoming
connections, show green dot if it happens, and red dot and error message
if not.