Conditions for which the Hilbert transform returns a correct phase
A single instantaneous phase estimate may or may not make any sense if there is more than one frequency peak in the signal's local spectrum.  So, to get a better single frequency and phase estimate, you may want to first high-pass filter your signal to remove the spectrum of any D.C. bias or slow trend, and also band-pass filter the signal to remove any outside spectral peaks due to noise or interfering signals.

So my off-hand guess for a condition is that the local spectrum has a single clear frequency peak with a monotonicly decreasing spectral magnitude away from that one peak.  All the way to zero at DC for real signals.