Rietveld (FullProf) general background function
If I recall correctly, Chebyshev polynomials are preferred over simple polynomials because of their termination behavior (i.e. they tend not to diverge wildly), although I don't have a reference for this handy.

Dinnebier's admonition to use higher-order polynomials with care is effectively a warning not to over parameterize your fit. If you inspect the two fits with 6- and 24-term backgrounds, my guess is that you will find the over parameterized version nudged your Bragg peaks up or down as necessary to better match your observed data. Hence, your residual has diminished but your fit is not more meaningful.

An overfitted background precludes accurate extraction of the observed Bragg intensities, leading to correlated errors in related factors (e.g. occupancy, position, etc.).