In Welch or Bartlett method, original signal is divided into L segments, of D overlapping (D=0 in Bartlett method). Then FFT is done for each segment, and the result is the average of all segments' FFTs. Averaging reduces the variance. But apart from this effect, is there anything else to take into account when choosing the number of segments for a given signal?
If you increase the number of segments for a fixed total signal length, each segment becomes shorter. This means that you'll have a trade-off between the estimate's variance and the frequency resolution. Averaging many segments will reduce the variance but by using shorter windows, the ability to resolve closely spaced narrow band components is decreased. Shorter windows also decrease the peak amplitudes of any narrow band component. Shorter windows basically average out any details in the spectrum.