The most I've been able to get into one race at a time is 4 (most of the endurance races -- I'm up to 24 Hour LeMans and it only allows 3).
The drivers switch off during the race at each pit stop, so it's not that you will be having multiple cars on the track at once.
The point of having more than one is that the drivers get worn out or moody (the arrow next to their name) and it's sometimes better to pick one that isn't on a down arrow.
But really you only need one up until the endurance races (and even then you only NEED one).
Also, once your driver's Experience/Career bar gets full their skill will start declining, and it's time to retire them.
The good news is that in the endurance races they can get very high levels in very few races. I have one guy that's level 31 with only 11 races under his belt.