Until 1974, the two races weren't held on the same day, so it would have been impossible.
From 1974 to 1992, they were held on the same day at the same time. So still impossible.
From 1993 to 2004, the race in Charlotte was scheduled in the evening, with the one in Indianapolis starting at noon. This technically allowed drivers to compete in both events (with a pretty tight scheduling window) by completing the Indy 500, taking a helicopter to the airport, flying to an airport near Charlotte, and taking another helicopter to that race.
John Andretti, Tony Stewart, and Robby Gordon attempted this. Only Stewart actually managed to complete the full distance of both races (in 2001).
In 2005, Indy moved the start of their race to 1 PM. Since the schedule was already very tight, this made it effectively impossible to compete in both races again.
However, starting in 2011, the starting time of Indy has been moved back to noon. I don't believe anyone has managed to qualify in both races yet since the change (as of 2014).
NASCAR
The Indy 500
None, Jr races NASCAR not Indy.
A lot less than a Nascar
yes
Indy 500, open wheel cars, NASCAR holds the brickyard 400 at Indy every year.
33
2007 winner Dario Franchitti, now racing in NASCAR, did not race the 2008 Indy 500.
Oxygen
no
Willie T. Ribs
The IRL, Indy Racing League - NASCAR and Motorcycles... Community Day is GREAT!