It depends on so many factors, including your native language, prior experience learning languages, belief that you can become fluent, willingness to spend free time with the language, the amount of time you can dedicate to English in those three months, and the intensity of your interest in movies, books, podcasts, videos and speakers of English. It also depends on your definition of fluency.
Some people see fluency as nothing less than perfection in every situation imaginable in the language. Others lower the bar so that whatever level they will have reached in 3 months counts as fluent. We will always quibble about definitions, but fluency can be thought of as the level at which you can have meaningful conversations with a native speaker without strain for either of you. You don't struggle to understand their speech at a normal pace. You don't constantly ask them to repeat or rephrase what they said. They don't feel they have to simplify their speech or slow down. They don't struggle to understand you. You can express what you want without feeling restricted because you don't know how to formulate the thought. Most importantly, it flows. The correct words and arrangement tumble from your lips without much mental strain. This is probably between a B2 and C1 level, which is no small accomplishment.
Therefore, I think some people can do it in three months, especially when you consider that many non-native speakers of English have at least some background in the language due to years of schooling. I think in most cases, though, it's not a realistic goal because most people don't have the time or the interest. You're going to need hundreds of hours to reach a true B2/C1.
One famous proponent of the "fluent in 3 months" approach made grandiose promises about becoming fluent in a Sinitic language in three months and did not reach the goal, despite putting in hours a day. The lesson? Don't make people think it's realistic for just ANYONE to be fluent in three months.