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
The number 40 car has been driven by several drivers in the Indianapolis 500 throughout its history. Notably, drivers like Johnny Rutherford, who won the race in 1974, and Tony Stewart, who competed in the number 40 car during his Indy 500 appearances, are among the most recognized. The car number has seen various teams and drivers over the years, reflecting the changing landscape of the event.
yes
33
Indy 500, open wheel cars, NASCAR holds the brickyard 400 at Indy every year.
2007 winner Dario Franchitti, now racing in NASCAR, did not race the 2008 Indy 500.
Oxygen
Willie T. Ribs
no