I start assessing a game immediately upon launching it. For example, I hate sitting through splash screens, and if you make me do that, I will get annoyed each time. I don't like being annoyed with games, because they're supposed to be a fun and relaxing experience.
When it comes to other aspects of the game that may be more game-related, those are quite variable. For example, I may just not enjoy the way a platformer feels, and I can get that within the first few minutes. I may not enjoy the way combat feels in a game as it's too slow and clunky compared to what I'm expecting. Even if it gets better later in the game due to unlocked combos, etc., I don't want to play through (potential) hours of boredom just to have a little fun. I'll find a game that keeps me entertained all the time.
Honestly, I don't think that you're really making a strong case for the games to which you are referring. If a game starts off very boring (for whatever reason), then it isn't doing its job! A game is supposed to keep the user engaged through one of its aspects (gameplay, story, etc.), but if a game is boring, then it isn't accomplishing that goal. You mentioned Persona 4 (in a portion that I cut out of your quote), and I think that's a great example of this. I bought Persona 4 Golden awhile ago, and I had a hell of a time trying to play it. The problem was... the game was just too boring up until you actually start fighting (it was somewhat more interesting when you first fall into the TV). I think it took me about an hour and 15 minutes to hit that first part and I've finished the portion where you start fighting and I'm at about two hours. What made it really boring to me is that it's just full of normal day-to-day sort of things, which means there's very little character development and minor bits of story introduction.
I recently read through the Final Fantasy IV translation comparison, so that game is a bit fresh on my mind. It probably takes about 15 minutes to hit the prologue for that game (when you leave Baron Castle with Kain), which is an hour less than Persona 4, but I'd argue that it accomplishes just as much. Ni no Kuni is a recent JRPG entry, and it takes... about 10-15 minutes to get you to the other world, which includes quite a bit of story elements.