In Star Trek: First Contact, Picard and Worf have an argument leading to Picard calling Worf a coward. After he comes to his senses, he says:
Picard: Mr Worf, I regret some of the things I said...
Mr Worf isn't too pleased, and Picard makes a noncommittal backtrack, saying that Worf is the bravest person he's ever met. However, that "some" seemed to bother Worf.
What did Picard mean, and what did he not regret?
Here's the transcript of the relevant part of the scene:
[Enterprise-E bridge]
CRUSHER: So much for the Enterprise-E.
PICARD: We barely knew her.
CRUSHER: Think they'll build another one?
PICARD: Plenty of letters left in the alphabet. ...Mister Worf. I regret some of the things I said to you earlier.
WORF: Some?
PICARD: As a matter of fact I think you're the bravest man I have ever known.
WORF: Thank you, sir.
PICARD: See you on Gravett Island.
Here is the video on youtube
To me, it's a rather obvious example where the arrogant side of his is coming out: He rather downplays the impact and magnitude of his words, and Worf calls him out on it.
Because if there is one negative thing that we can say about Picard, then that at some times, his pride, or arrogance, gets in his way. And this is one of the times. In fact, you could argue that it's his own pride (in always doing The Right Thing) that blinds him to the fact that he is acting out of vengeance, and not rationally, during this exchange in the same movie:
LILY: Jean-Luc, blow up the damn ship!
PICARD: No! ...No!
(Picard breaks the starship display cabinet with his phaser rifle)
PICARD: No! ...I will not sacrifice the Enterprise. We've made too many compromises already. Too many retreats. They invade our space and we fall back. They assimilate entire worlds, and we fall back. Not again! The line must be drawn here, ...this far, no further! And I will make them pay for what they've done.
LILY: You broke your little ships. ...See you around, Ahab.
This is not something invented for the movie, though. This is discussed in Star Trek Picard between him and Admiral Clancy. Even though we, as the audience, are obviously on Picard's side, we can see how Clancy perceives him:
Picard: The Federation does not get to decide if a species lives or dies!
Clancy: Yes we do. We absolutely do. Thousands of other species depend upon us for unity, for cohesion. We didn't have enough ships left. We had to make choices, but the great Captain Picard didn't like his orders.
Now, you could still argue that Picard is right, Clancy is wrong, therefore what she perceives as arrogance is in fact being noble.
But there are several other occurences where Picard is arrogant and doesn't even notice it. Just remember how blind he was to Wesley's obvious talent, just because he didn't like him on the bridge, in Encounter at Farpoint:
(Wesley steps onto the Bridge)
PICARD: But don't touch anything! Try it out. (the captain's chair) The panel on your right is for log entries, library computer access and retrieval, viewscreen control, intercoms, and so on. Here we have
WESLEY: And here, the backup conn and ops panels, plus shield and armoury controls.
PICARD: The forward viewscreen is controlled from the ops position there
WESLEY: Which uses high resolution, multi-spectral imaging sensor systems
PICARD: How the hell do you know that, boy?
WESLEY: Perimeter alert, Captain!
CRUSHER: Wesley!
WESLEY: I'm sorry.
CRUSHER: You shouldn't have touched anything
PICARD: Off the bridge! Both of you.
WORF: You have a perimeter alert, sir.
CRUSHER: As my son tried to tell you!
(Crusher and Wesley step back into the turbolift)
Perhaps the most intimate insight into the negative traits of his personality, as perceived by others, can be seen in TNG's Episode "Family", where his nephew tells him how his father thinks about Picard:
RENE: Why have you been away so long?
PICARD: Well, Starfleet keeps me very busy.
RENE: Father says you don't like it here.
PICARD: I'm sure you misunderstood.
RENE: No, I didn't. He said so.
PICARD: Well, Robert and I, we. Perhaps it's time to change all that.
RENE: You know, you don't seem so arrow. Arrow. You know.
PICARD: Arrogant?
RENE: Yes, arrogant. You don't seem that way to me. What does it mean anyway, arrogant son of a...
PICARD: Let's talk about that later, shall we?
Personally I think that Picard regretted everything that he said which made him too ashamed to even admit that he had regrets because he should have known better. Picard is a great captain, but let's face it, he is not flawless and his arrogance sometimes get's the better of him (something that Q noted a few times over the course of the series). And something that is also noted by Lily, who criticized Picard's beliefs of them being "more evolved" than humanity of her time while still being little better than Captain Ahab from Moby Dick. This conversation took place before Picard's apology to Worf.
