The standard picture of the neutron star core has coexisting neutron and proton superfluids, with the proton component forming a type II superconductor threaded by flux tubes. I show that this picture is inconsistent with observations of long-period (~1 yr) precession in isolated pulsars. Hence, either the two superfluids coexist nowhere in the stellar core, or the core is a type I superconductor rather than type II. As I will discuss, either possibility would have interesting implications for neutron star cooling and theories of spin jumps (glitches).