Can two aircraft be allowed to stay on the same runway at the same time?
There is another scenario other than "line up and wait" that the other two answers cover...

A "flight" of more than one aircraft may be cleared as a group to take-off. This can consist of many aircraft sharing the runway during take-off (and even landing). 

So you can have multiple aircraft on the runway when:


A "line up and wait" order is issued to line up on the runway after another aircraft takes-off or lands. This is usually done so that the departing aircraft can get prepared while waiting for wake-turbulence to clear.
ATC clears multiple aircraft as a "flight". The aircraft must be part of a coordinated group and a single aircraft is the "lead" aircraft for the flight that receives/coordinates with ATC for clearances. The entire flight group will enter the runway and take off in a pre-coordinated fashion.


There are specialized scenarios as well, for example during Airventure Oshkosh. Aircraft will be required to "land on the dot" and multiple aircraft may be on the runway for take-off at the same time, although ATC coordinates them so that they don't depart at the same time, they are asked to line up "on a dot" (a colored circle on the runway) and will then be cleared to depart from there.

As an extreme example:


                
            

116 aircraft lined up on the runway and cleared for take-off at the same time.