What kind of periodic signals cannot be represented with the Fourier Series?
Oppenheim et al. give an answer on page 196. Paraphrased:

Define the approximation error as:

$l_N(t) = x(t) - \sum_{k=-N}^{k=+N} a_ke^{jk\omega_0t}$

There exist a class of signals for which:

$lim_{N\rightarrow \infty} l_N(t) \neq 0$

For these signals, the Fourier Series is, at best, an approximation.

Oppenheim et al. reference R. V. Churchill Fourier Series and Boundary Value Problems.