I have an E61 on v2.xx, not an E65, but here's my experience with how it SHOULD work I believe. When I first got the phone, I set up my IMAP service on the phone and experimented with it. I found that once its all set up, I go in and set the account to "AUTO RETRIEVE" every 30 minutes. Once I set that setting in the phone, after about five or seven seconds, it makes a GPRS connection on its own (I know because of the little symbols under the signal strength bars). Once it connects, it STAYS connected to GPRS. Anytime I get an email, the email header automatically pops up in my box (within about 15-20 seconds or so). Every NOW AND THEN, something goes awry and it would DISCONNECT from GPRS service. SOME of those times, I would still keep getting emails but OTHER times it really wasn't talking to any IMAP server (I know this because I don't get alerted to new emails...).
All that describes what used to happen with my v2.xx firmware. Once I upgraded to version 3.xx, I haven't had a single issue at ALL with IMAP. Once I connect, the phone stays on GPRS and I have "push" working perfectly. A neat trick with the S60 IDLE settings that I use is to be able to "turn off" IMAP-IDLE. If you go into Email Settings and change the AUTO RETRIEVAL setting to NEVER, then IMAP-IDLE is effectlvely "off". This way, at night when I'm sleeping, I am not bugged my emails... then, when I wake up, I simply turn on "AUTO RETRIEVAL" to "ALWAYS" and my phones message box re-syncs w/ the server. The only downfall with IMAP-IDLE that I've found is that my battery doesn't last as long with a constant GPRS connection. If I leave off IM, Email IDLE and don't browse the internet, I could probably get at least 2-3 FULL days... maybe more. Yesterday, I turned my phone on, fully charged at 9am. I used it to browse CONSTANTLY, had IDLE activated all day, was on IM+ all day, used Google Maps all day and probably sent 200 text messages.... it finally started beeping battery low at 4am ...19 hours. I love my E-Series.
All in all, I would think that the E65 would have the same implementation of IMAP. Here's a way to check. When you lose your "EDGE" connection (GPRS), send an email to your IMAP account ... if you have auto retrieve ON then the emails should come through to your phone. If they do, then maybe IMAP-IDLE IS, in fact, working, but without the GPRS indicator. Otherwise, there may be a problem with 1) your IMAP account, 2) your provider's data services, or 3) your (relatively new) firmware.