Just thinking out loud here.
Another negative effect that increasing cap values can have is on the PSU's stability. I know we're dealing with Class A small signal circuits and this really comes into play with Class AB operation tube power amps, but I thought it still worth mentioning because it's yet another reason why blindly increasing the filter capacitance in vintage tube gear just for the sake of increasing it can be a bad idea. I realize that not a lot of vintage push pull amps have CLC filters before the power tubes, but some do (and I've DIY'ed several) and some of the below principles do apply to Class A in a certain measure as well.
Consider a sudden, high amplitude transient at low frequencies, where the CLC pi filter must instantaneously deliver a lot of current. A small value input filter cap naturally has more voltage sag but recovers from transients quickly and smoothly, while a large value input cap exhibits better regulation, but upon recovery its output voltage can significantly overshoot the nominal rail voltage and then settle back down with ringing oscillations. The choke's DCR will help damp the ringing, but of course Class AB amps with CLC filters don't ever (or rather shouldn't) have high DCR chokes because of the resulting poor voltage regulation. Unless of course, it's a guitar amp where serious voltage sag is what you're actually looking for.
Increasing the value of the CLC output cap reduces ripple, improves regulation, damps oscillations, and lowers the PSU's impedance at low frequencies. But, it can be a two-edged sword due to the bigger cap's typically higher ESR, which means higher PSU impedance at high frequencies, especially as the cap ages. Ringing will also be pushed to a lower frequency because the larger cap changes the time constant of the RLC lowpass, which depending on the amplifier topology and AF frequencies of interest might or might not have a negative effect. Lots of stuff to consider....
As several have mentioned, large value caps after tube rectifiers can send them to an early grave due to the increase in RMS current value. Abbey noted that it also stresses power transformers, and not only does it increase the I²R heat gain in the secondary, but if it actually saturates the core on transient peaks you'll then have harmonics of the AC line frequency in your power rail.
FWIW, the many times I've tried greatly increasing the capacitance value of a C filter or the output of a CLC beyond "normal" values (in both Class A and Class AB circuits) has not once yielded positive sonic results, at least not to my particular set of ears. The almost universal negative artifact has been a recessed, lackluster midrange, and bypassing the big caps with small value film caps to cut the ESR has always introduced other sonic anomalies.
Just some food for thought, so to speak.
Rusan