Q:

Why wasn't Captain Picard ever promoted beyond Captain?

At the end of Nemesis movie, Captain Picard is still captain despite having served for many years.
Let's go over some key years from his career as provided by StarTrek.com.

Year 2333: Assigned as commander and first officer on U.S.S. Stargazer.
Year 2355: Promoted to captain of Stargazer after death of his superior in battle.
Year 2363: (or 2364?) Chosen to command relatively new Galaxy-class U.S.S. Enterprise-D.
Year 2372: Chosen to command relatively new Sovereign-class U.S.S. Enterprise-E.

He had a 22-year career as first officer and later captain of the U.S.S. Stargazer with an even more impressive record as captain of the fleet's former flagship U.S.S. Enterprise.
Like an admiral, he has led multiple ships several times. He's comfortable handling sensitive diplomatic negotiations like a diplomat. The man should have been promoted to admiral or appointed as ambassador. Yet, he's still just a captain.
He was offered a promotion to admiral if he would become the commandant of Starfleet Academy, but he turned that down. (I think that was in the second season of ST:TNG.)
Others have been promoted faster than him. Riker became captain after spending less time as Commander than Picard. Captain Janeway became admiral after only 7 years as captain (years 2371 to 2378).
Why hasn't Picard been promoted yet?

Did he annoy some superior officers?
Was he held back in his career?
Did he choose to stay a captain?
Was it because Starfleet lost many good captains in the battles against the Borg? (You could say Starfleet could not afford to promote him because they needed good captains after the two Borg invasions.)
Was it because Starfleet wasn't sure about him after his assimilation? (It was mentioned in the First Contact movie they had some reservations about him going against the Borg.)

A:

Because he wanted to stay a captain. As you said, he is offered a promotion in TNG: Coming Of Age, and he declines it for this very reason:

QUINN: That's not enough. I want to promote you to Admiral, and I want you to take over as Commandant of Starfleet Academy. 
  PICARD: The Academy.  
  QUINN: Yes.  
  PICARD: The Academy?  
  QUINN: I need you close.  
  PICARD: Then there was never a problem with the Enterprise.  
  QUINN: No, but I had to be sure you hadn't been co-opted.  
  PICARD: Greg, this is politics, and I'm not good at politics. Surely there are others who are better suited. 

Later in the episode:

QUINN: Wish I could convince you to change your mind. 
  PICARD: I'll serve you better here.

Later, Captain Kirk directly told him not to ever accept promotion in Star Trek: Generations, further affirming his position.

KIRK: You know, maybe this is less about an empty house than that empty chair on the bridge of the Enterprise. Ever since I left Starfleet I haven't made a difference. ...Captain of the Enterprise, huh? 
  PICARD: That's right. 
  KIRK: Close to retirement? 
  PICARD: I'm not planning on it. 
  KIRK: Let me tell you something. Don't! Don't let them promote you. Don't let them transfer you. Don't let them do anything that takes you off the bridge of that ship, because while you're there, you can make a difference.

(Emphasis mine)
Kirk later reinforces his philosophy of "making a difference" with his dying words:

KIRK: Did we do it? ...Did we make a difference? 
  PICARD: Oh, yes, we made a difference. Thank you. 
  KIRK: It's the least I could do for the Captain of the Enterprise. ...It was ...fun. ...Oh my.

Ultimately, Picard wants to be an explorer, not an administrator, and while he is undoubtedly an accomplished diplomat, he is not interested in the mundane pleasantries of administrative diplomacy. We see this attitude expressed in Star Trek: Insurrection.
After a long conversation about traditional greetings and mission rescheduling:

RIKER: The diplomatic corps is busy with Dominion negotiations. 
  PICARD: Oh, so they need us to put out one more brush fire. Can anyone remember when we used to be explorers?

