Yes. The Doctor, for example, usually suffers some regeneration trauma -- generally in the form of short-term mental instability -- but has sometimes suffered much more serious effects. The Fourth Doctor's "nuttiness" early in "Robot" was one instance of the milder form & the Sixth Doctor's aggression early in "The Twin Dilemma" was another. Notably, both of these examples were in the form of an exaggerated version of that incarnation's regular personality. The Fifth Doctor's collapse in "Castrovalva" (which required him to resort to the Zero Room) & the Tenth Doctor's collapse during most of "The Christmas Invasion" were instances of more serious trauma.
The Time Lord eventually known as "the Eleven" developed a severe form of multiple personality disorder as a result of a regeneration going wrong, as revealed in "The Eleven" (audio story).
Additionally, in "Trial of the Valeyard" (Big Finish audio story), the Valeyard claimed to know of Time Lords who had regenerated into Time Tots (Time Lord children), Time Lords whose bodies had changed but personality had not (and vice-versa), Time Lords who had regenerated inside out, and Time Lord who had regenerated into horrific non-humanoid forms, amongst other oddities. The story left it ambiguous as to whether or not he was telling the truth.
It's also possible for a regeneration to be unfinished which as evidenced in "The Christmas Invasion" would result in a horrible glowing monstrosity flickering in and out of regeneration.