Observations have suggested that some low-mass stars have larger radii than predicted by 1D structure models. Some theoretical models have invoked very strong interior magnetic fields (of order 1 MG or more) as a possible cause of such large radii. Whether fields of that strength could in principle be generated by dynamo action in these objects is unclear, and we do not address the matter directly. Instead, we examine whether such fields could remain in the interior of a low-mass object for a significant amount of time, and whether they would have any other obvious signatures. First, we estimate the timescales for the loss of strong fields by magnetic buoyancy instabilities. We consider a range of field strengths and simple morphologies, including both idealized flux tubes and smooth layers of field. We confirm some of our analytical estimates using thin flux tube magnetohydrodynamic simulations of the rise of buoyant fields in a fully convective M-dwarf. Separately, we consider the Ohmic dissipation of such fields. We find that dissipation provides a complementary constraint to buoyancy: while small-scale, fibril fields might be regenerated faster than they rise, the dissipative heating associated with such fields would in some cases greatly exceed the luminosity of the star. We show how these constraints combine to yield limits on the internal field strength and morphology in low-mass stars. In particular, we find that for stars of 0.3 solar masses, no fields in flux tubes stronger than about 800 kG are simultaneously consistent with both constraints.