They're pretty locked into a sound, which I happen to really like. I think part of what limits them is their use of what they call the "basitar" and "guitbass" which is explained on their Wikipedia page:

Ballew and Dederer/McKeag play a basitar and guitbass, which are regular, six-string guitars with special modifications: Ballew's instrument has two bass strings (with which he plays bass parts), and Dederer/McKeag's instrument has three guitar strings (used to play guitar parts). The original idea came from Morphine frontman Mark Sandman, with whom Ballew had previously worked.

Guitars modified in this way use heavy-gauge strings for a heavier sound and are normally tuned in Drop D, though the Presidents play half a step lower in C#. For a basitar, the strings are placed in the D and G positions. Chris uses a .60 gauge string tuned to C# and a .36 tuned to G#. On a guitbass, the strings are placed in the A, D, and G positions. .54 gauge tuned to C#, .42 to G#, and a .32 gauge tuned to C#.

In a way I like that they have kind of a singular, self-cohesive vision of rock music. On Freaked Out and Small they do play around with their sound a bit, for example in Tiger Bomb (live version). They also play with normal bass and guitars on that album.