Password Incorrect and Cannot Get Mail Popups on iPhone 4

Device: iPhone 4
Carrier: Verizon Wireless
Country / Language: USA/English
OS / Browser / build number (if applicable): Version 4.2.6 (8E200)
Ive never needed to post to a forum before because Ive always been able to find the answers needed before, but after HOURS (10+ at least) of reading (and re-reading) posts, I havent found an answer that works and it appears that many of those who seemed to have had similar problems were either human error or fixed by other remedies. I guess its time for me to start contributing:
The problem:
Just got an iPhone 4 and read the instructions on how to properly set up Gmail using a Microsoft Exchange account so that my contacts, mail, and calendars sync with the phone. See the following website:
http://www.google.com/support/mobile/bin/answer.py?answer=138740&topic=14252
Upon completing the instructions, my contacts, emails, and calendar DID properly sync. Within 24 hours of this correctly working, I began having popups (or whatever theyre called in Apple land) saying the following:

Password Incorrect
Enter the password for the Exchange account (email account name)
After correctly entering the password multiple times (at least 3, but maybe more) I was shown a new popup message:
Cannot Get Mail
The user name or password for (email account name) is incorrect
It should be noted that at no point since the accounts correctly synced to the iPhone was any password changed. Some unknown number of hours later, I begin receiving push notifications for email from these accounts on the iPhone. (The problem had seemingly fixed itself.) The emails had been sent a great deal of time prior to the notifications and when I checked my email from the phone, neither popup was displayed and I was able to both send and receive emails from the phone.
Fast forward a month later and these events are nearly a daily occurrence and each time it fixes itself sometimes minutes, sometimes hours later. My frustration levels through the roof!
These are some (although Im sure Ive forgotten some) of the things Ive read about and tried with no success:
1. Verified user names and passwords - Ive checked and double-checked all user names and passwords on both my phone and my computer. All user names and passwords are correct.
2. Re-install account - All accounts have been deleted and installed again (3 separate times) using the instructions found by following the link I posted above. Ive also tried variations on the install instructions at the suggestion of others: user name only (instead of @gmail.com) using @googlemail.com
3. Restore phone - Hooked up the phone to my computer and did a full-phone restore back to the way it came out of the box and started again (4th install of accounts)
4. Changed passwords - Possible problem with the strength of my password? (It was suggested, so I tried it)
5. Verified the IMAP was enabled in Gmail. (It is.)
6. Unlocked Captcha - I have done this multiple times both via my computer AND the web browser on the iPhone.
https://www.google.com/accounts/UnlockCaptcha?
There are some random observations Ive noticed about this problem. They may not be anything, but maybe itll make sense to someone else:
1. This problem seems to happen more frequently (although certainly not always) in conjunction with the turning on and off of Airplane Mode (I travel a lot).
2. There doesnt seem to be a pattern in the length of time between when the problem starts and when it seemingly corrects itself, nor the time between then and when it starts again.
3. I have 3 Gmail accounts using this phone:
Acct A is set to sync mail and contacts ONLY
Acct B is set to sync mail and calendar ONLY
Acct C is set to sync mail ONLY
Ive noticed that Acct C seems to have the problem almost never compared to the other two (which always have the issue at the same time). Ive also tried having all three accounts sync all things, but that didnt help either.
If anyone can think of anything else, PLEASE let me know! I know as soon as I click Post question Ill think of something else I tried, but Ill update it if I think of something. If Apple and Gmail cant work this out, Im going to have to pick one because this is unacceptable...
iPhone 4 iOS 4