There may be many different places from which you could commence an instrument approach (called the IAF, Initial Approach Fix). The pilot can negotiate with ATC for which one to use, or ATC may dictate which one you will use. As well as ATC may give you vectors to the final approach course which means you don't need any IAF.
ATC should never clear more than one airplane for an approach in such a way that aircraft would come into conflict when initiating an approach.
Example, suppose 2 planes are heading to podunk airport and ATC cannot provide vectors to the final, then ATC will clear one plane for the approach and the other plane will get to hold somewhere until the first guy has cleared the area.
Many of the non-RNAV/GPS approaches begin with a track away from the airport and then a procedure turn back onto the final approach course. This means if there were multiple planes on the approach that some might be heading straight at each other.
When ATC is giving vectors to the final, they can stack multiple planes in trail on the same approach, with the assumption that no aircraft will ever be heading at each other.