Pound & Rebka's conclusions are questionable. They fired photons downward at an absorber, and found that the electrons didn't like absorbing the photons unless they were slightly redshifted. To get better absorption, Pound & Rebka had to move the emitter away from the absorber, producing a Doppler redshift. They concluded that the photons were getting blueshifted by gravity on the way down, and perfect absorption could only occur if the photons began with a slight redshift. Thus the necessary degree of redshifting could be a measure of the degree of blueshifting that was occurring in flight. So far so good.
But Pound & Rebka did not rule out other well-known possibilities, such as effects on the absorbers themselves caused by external fields.
For example, in an electric field, what would otherwise be a single absorption/emission line in a spectrum gets split into two lines. This is known as the Stark effect, and it's one of the primary methods for detecting electric fields from a distance. The reason for the split is that electrons changing shells move faster if they are going with the field, and slower if they are going against it. When emitting photons, the faster electrons emit blueshifted photons, and the slower ones emit redshifted photons. Since these could be traveling in any direction, an observer at a distance will get both varieties — thus the single emission line has been split into two lines. The corollary to this rule for emission is that during absorption, an electron that has to fight against the electric field can only absorb a redshifted photon, and an electron being accelerated by the field can only absorb a blueshifted photon.
For precisely the same reasons, we should expect spectral lines to be split by a gravity field — electrons moving upward/downward in the field should absorb only redshifted/blueshifted photons. But Pound & Rebka didn't look for spectral splitting. They tested for absorption of the redshifted downward photons, and confirmed their preformed conclusions by testing the absorption of blueshifted upward photons. To rule out spectral splitting because of an external field operating on the absorber (as with the Stark effect), they should have looked for both effects in both configurations. They should have known this, since the Stark effect was discovered in 1913, and they did their experiment in 1959. But none of the write-ups on this experiment mention this double-check, so we can only assume that it was neglected. As such, this is still an open issue.
Pound & Rebka didn't take field splitting into account.