There's no real rule determining when certain users may begin to fossilize. It varies widely by the individual and by the environment in which the language is learned.
Fossilization most often occurs in an inadequate learning environment. This usually means learning a language in a classroom, as opposed to learning it in the country where it is natively spoken. Many aspects of a language simply cannot be taught in a classroom, where one generally learns a highly academic version of the tongue, as opposed to the colloquial language.
However, fossilization can still occur despite complete immersion in a foreign language environment for decades, a well documented phenomenon amongst, for example, immigrants. Clearly, this is not the only issue at stake.
Fossilization often means that certain aspects of the language were learned incompletely or incorrectly, such as grammatical features like conjugating verbs in the wrong fashion or using the wrong vocabulary, in such a manner that they cannot be unlearned and replaced with correct usage.
Fossilization may also consist of a sort of subconscious clinging to aspects of the learner's mother tongue, for instance, with syntax and phonology. This may reflect an inability to similarly “unlearn” characteristics of a mother language for the purpose of learning another; the native language so deeply hardwired into the brain that its paradigms cannot be replaced when attempting to learn a new and foreign language.