Mail app asking for passwords it already knows, taking accounts offline, intermittently failing to connect IMAP/SMTP

Question

Recently I have been experiencing intermittent connection failures with Mail in OS X 10.7.5 on both our Mac Pro and MacBook Pro.
The little "lightning bolt" icon or the "exclamation point" icon will occasionally appear next to some or all of my accounts in the left column and Mail will (sometimes) prompt for passwords already in my keychain and then take the accounts offline. Sometimes the failures take place during IMAP logins and sometimes during SMTP sends. Sometimes they affect my iCloud accounts, sometimes my Gmail accounts, and sometimes both.
When I try to take the account(s) back online, Mail sometimes fails to do so. Quitting the Mail application completely and relaunching it will randomly result in the problem persisting or going away. What makes this so confusing is about half the time, Mail works just fine.
This is only happening on OS X, in Apple Mail, our iPad and iPhones appear unaffected even when running on the same router with the same ISP. I can also verify that all the iCloud and Gmail accounts in question can be reached successfully using other mail client software from Windows in my various virtual machines. Therefore I believe this is an application issue and not something related to my network.
When the issue is evident, Connection Doctor will fail on some or all counts...when it goes away Connection Doctor will show all green lights again.
This issue has only manifested in the past week or so, and both machines are up to date on Software Updates. Perhaps something pushed out recently from Apple is causing this? Is anyone else having a similar experience?

Answer 1

Same issue, too.  But I have nothing better to do than try to fix Apple problems that appear mysteriously and randomly; I'm sure you're the same way.
The REAL REASON, which can no longer be overlooked, is that Apple's (probably underpaid sweatshop Chinese) employees build toys for grown-ups and do little more than that.  I wish I had never bought my Mac (2009) since few things about Apples are "intuitive" unless you've been an Apple cultist since Jobs introduced the first Mac.  Oh, and because things mysteriously stop working, something I don't recall happening with any of the PCs I've owned. 
Apple's version of Excel is a total joke and on an update that I did not ask for, the "Save As" feature, a convention that was 30 years old and worked well, was removed in favor of "Duplicate, Rename, Save."  What 22 year-old Apple product manager was allowed to make this change? 
I foolishly compounded my Mac purchase error by buying an iPhone 4S.  On that lovely ($200 Verizon-subsidized) product, I can no longer discover my home network, rendering my ($100) Airplay for outdoor music useless.  (But I probably can post a picture directly to Facebook or something!  More toyish behavior in which Apple specializes.)  Can I get support on these POS with a phone call?  No, I need to rely on a Aapl cultist to devise some esoteric key combo or a "jailbreak" on a Support board or blog site.
I'm eligible for a phone upgrade on 8/13 and I'm counting the days.  Will upgrade to a good large-screen laptop asap as well and then throw the Mac out my office window.  Junk, just junk.  Apple's stock is at $433 from $700 and it's still overpriced.  How long will this rant last before it's sanitized by the Cupertino Big Brother? Maybe what the children who design product ought to do instead is BUILD BETTER STUFF.