Does the number of games that a player is involved in reduce
It can. I've been in situations where I had six or eight games all active at the same time, and a limited time slot in which to participate. And on days where I needed to respond to all the games, I definitely felt hurried and wrote more abbreviated versions of posts than I would have otherwise done. It wasn't always a problem...sometimes I had plenty of time, could sit down and figure out my posts...it was actually relatively rare that I needed to respond in all the games at the same time...and sometimes, a shorter post was actually the most fitting thing to post at the time. And if there are only three other players in a game, and all the posts are in the same thread, it's less of an issue than if you've got eight or nine active players and posts are scattered across four or five threads...or if each player has only posted once, versus two or three players having an extended conversation that needs to be read before a response is posted.
But, yes, it can impact the quality of the posts. How much of an impact is impossible to say, however, and if it's a player who's creating really good content, nobody may notice the difference without some serious analysis. As mentioned above, actual available time to post will have a greater impact...but if you've only got an hour of free time, splitting it four ways versus splitting it eight ways is a significant difference in the amount of time available for each post, especially when you factor in time lost in reading other posted material to catch up to where they need to be.