The simple answer is, Yes, You can buy a Mattress without buying a Box Spring. But, if you are using an Old Box Spring, this could cause damage to your mattress and in turn could cause a manufacturer to deny any warranty claims that may arise.
Another thing to take into account when buying a Mattress and Box Spring. The Boxspring is a small portion of the mattress set price so may be worth getting the set price. On average, the Boxspring will cost less than $200 and will eliminate the questions if you ever have a warranty claim.
When you are thinking of not buying a Boxspring, there is an instance that one is not required. If you have a platform bed that has a solid flat surface, a Boxspring is not required.
If you are buying a Mattress and Boxspring, weigh your options to see if it makes sense to buy a set or a Mattress by itself. Buying the Boxspring is not mandatory so don't let anyone tell you that you don't have an option, but it may make sense to buy a Mattress as a set.