I suppose that it depends on your definition of acceptable and your definition of cult.
Realistically, every religion began with a handful of people who believed the same thing, or had some sort of experience that had them believe that thing. No matter the origins, as the groups developed they went from cult to religion.
On the acceptable side, if you are talking about accepted and left alone by mainstream culture, definitely. If you mean acceptable, as in makes rational sense, not so much. At the core, each one is stating that it knows better than everyone else, based on no real facts or evidence. Just stories passed down over generations, with no real explanatory value. Even the fact that so many of them have come and gone speaks to that...
The definition of a "cult" is a small group of people who hold the same beliefs. All religions started as "cults". Christianity began as a persecuted cult in the Roman Empire. It now has over a billion and a half followers. So yes, a cult can definitely become very "mainstream".
The term "cult" is a relative word. It depends on where you are and what you believe. If you have beliefs that are very close to the Jehovah's Witnesses, for example, you would not consider them to be a cult. You would probably consider them to be a religion. If your beliefs are quite far from theirs, you could easily label them as a cult.
Yes, I believe a cult can become a somewhat acceptable religion, if it has enough members, financial backing, and media attention. Many people liken the Church of Scientology to a cult and it has become remarkably popular as an accepted religion, although it seems to cater to the more affluent.
I do not believe a cult can become an acceptable real religion. That doesn't mean it doesn't happen under the laws of the United States government or other governments. Religion is a tricky subject because, as a country, we want to provide religious freedom, but that also requires a very open mind to accept different religions. Cults probably don't make the cut, but if they have followers and meet the classifications of religion, then they deserve protection. So, it really depends on how a person decides rather something is acceptable or not.
I do not think that a cult can ever be an accept religion. I think that some cults are just too far out there that some people will not ever accept them for being anything more than a place where some people just follow a strange belief. I think that people should respect people's rights though.