I have noticed that some rifles, for whatever reason, do not have very tight bolts. There is quite a bit of "slop" when the bolt is in the open position. I have also seen a few that when the bolt has is closed and cocked, there is side to side and top to bottom movement, but no movement back in forth. I've seen it in Remingtons, Pre 64 Winchester, and Mausers. It seems more common in Mausers, especially unaltered military rifles.

My question is, does this affect accuracy? Are some rifles designed to have looser tolerances for a reason?

When the bolt is open it has little support and will flop around. Only the slop that is present when the bolt is closed affects accuracy, and by varying degrees.