I'm A Banana
Potassium-Rich
Speaking as one of THOSE GMs, the kind who fret about verisimilitude in the game-world, the kind who have plots moving in the background of their campaigns, yes, it matters, quite a bit.
I dunno, my players advance very fast (usually one sesson = one level) and I've never had a problem with verisimilitude. So I don't think there is any nessecary correlation between verisimilitude and the speed of level advancement.
I think it may also be an issue for GMs who don't like the idea of "killing stuff = experience points," including GMs such as myself who prefer the more leisurely and less bloodthirsty pace offered by story awards.
Story awards are all I give out, and my PC's go up in level VERY fast, so I don't think there is any nessecary correlation between story awards and a "leisurely and less bloodthirsty pace," either.
There is also no correlation between XP awards and in-came chronology. I can give my PC's one level/session in a 3 month campaign, but have it cover only one night of time in the world that they exist in. Similarly, even though only a week passes between sessions, it could be that decades passed between events in the game, during which the PC's were all at the same level.
So I guess just because someone plays different than you, that doesn't make their pace bloodthirsty and it doesn't make their world lack verisimilitude. Furthermore, a fast advancement in XP is not inherently related to an advancement in chronology.
The way it is spelled out by default, with 3-4 encoutners/day, and roughly 13 encounters/level, it takes an average person 4 (ish) days to advance one level, during which time he's doing nothing but fighting well-matched enemies. In-game, this means that they gain about 5 levels/month, and that someone who had to constantly fight evenly matched characters (even most adventurers don't do that...). But assuming these optimum conditions, a level 1 commoner could go to a level 20 in one season.
I think the true stretch of verisimilitude is that you wake up every day and fight 3-4 encounters that are evenly matched for you. EG: it's not in the XP, it's in the way it's handed out by the DM.