I know four people pretty well who have experience (direct-- either self, child, or siblings) with multiple grade skips of one sort or another.
Those who have
negative reports on it:
a) tend to be those who did multiple skips AND had another confounding variable in the mix
b) did NOT have the support of family/teacher
c) were hot-housed and gradeskipped as a means of managing a very poor quality school environment.
I know one person who was 2x accelerated and apparently bitterly regrets it, but this person is (reportedly) an Aspie who had unrecognized (?)issues aplenty that remained un-addressed both before and after the skips. I hardly think that counts as much of an indictment of acceleration, in light of the probable origins of the social problems that the person experienced.
I take that as a cautionary tale about kids who have extreme problems with social fit among agemates, and how those problems MAY be about poor intellectual fit, all right, but they may not go away with peers who are simply "older" either. I would probably be somewhat reluctant to skip a child who had a lot emotional regulation difficulty or social problems, myself... but in any event, being aware and managing those other related matters is probably the answer, not avoiding acceleration.
One of the major issues (authentically so, it seems) is about what happens during and immediately after adolescence-- a year of life experience and maturity DO seem to matter all of a sudden then, even when they haven't previously. It can also exacerbate problems associated with multipotentiality and major life-decisions which are often expected of kids in their last years of secondary education or first years in collegiate environs.
My best friend was another 2x skip-- but from sophomore to college, not as a young child. She had no difficulty at all adjusting to collegiate life-- BUT I'd consider her an exceptionally good candidate given that she was: a) highly self-motivated, b) driven, and c) unusually decisive about her life goals, even relative to many of her college peers.
One of my advisees as a young faculty member was also a 2X skip. He was slightly
emotionally immature relative to his classmates, but it wasn't noticeable other than that he didn't really share the aging out of teen interests with them. He was slightly out of synch as a college student, in other words. He very definitely wanted a more nurturing kind of relationship with faculty than is generally found on college campuses, however... but he wasn't a complete pest about it, either. He was fairly successful; though he had some trouble
emotionally managing irritations/upsets/setbacks, he was well within the norms for the population. I knew this student for two years before I learned that he was only 18 (and not a somewhat sheltered 20, as I'd thought).
Of the SINGLE skips that I've been party to, I can't think of one that hasn't gone well, either contemporary or of the 1970's-1990's vintage. My dd has two good friends who are both grade-skipped a single year, and both of those friends are not having any difficulty as a result of the skip.
One older classmate of hers
was disappointed that he had to make plans for an unexpected gap year given that West Point would not allow him matriculation as a 16yo cadet.

But that seems minor, as they deferred his enrollment a year.