Leave it a while and it will probably work. Then stop working again, then start again. I have the same issue, often supposedly down to using the wrong password. But then miraculously it works again using the same password (that is entered automatically).
If you have the same problem as me it's down to Bt, not the ipad. Having spoken to someone at bt at length the only think they could suggest was resetting the password across all my devices. That worked for a while but then the problems came back. I think it's BT's security software getting concerned about the account being accessed from multiple devices and assuming some of them are dodgy, so blocking them for a while. The BT guy alluded to this sort of, but couldn't suggest a permanent fix.
Bottom line is I'm having to live with it. Very frustrating, my only solution was to download their Mail App, which to be fair is good. But that means I sometimes have mail coming into two clients, which defeats the object somewhat.