There is not a specific time limit at which one or the other tense kicks in: the distinction is rather given by whether the action you are describing, and all its effects, are concluded, in which case you use passato remoto (for instance, “Giovanni Boccaccio visse nel XIV secolo”); or it happening or beginning in the past, but its effect being still with us, in which case you use passato prossimo (e.g., “In un certo senso la società capitalistica di tipo moderno ha avuto inizio nel XIV secolo”).
So, there is not a rigid distinction; the fact itself you use one tense rather than the other might mean you intend to convey a sense of distance or, on the contrary, an involvement in what you are telling.
In other, Luca Serianni's, words, «[il passato remoto] inserisce l'azione entro coordinate temporali nette, marcandone la compiutezza, lo stacco rispetto al presente», while a sentence expressed with passato prossimo «rivive il processo nei suoi riflessi successivi, collegando il fatto [...] con un implicito risultato attuale» (Luca Serianni, Italiano, XI, 377).
To make the point clearer, and taking an example from Serianni's passage quoted above, you may use any of the two tenses in the same sentence, according to what you mean: if you say «da giovane lessi molto» you might imply now you don't read anymore and don't consider that as a true part of your education; if you say «da giovane ho letto molto», you might want to elaborate on this: for instance, «e oggi mi considero istruito» or «e non mi lascio attrarre dall'ultimo premio letterario».
The sense that passato remoto refers to events far in the past while passato prossimo refers to recent ones is a consequence of the above. Events of one thousand years ago are often less relevant to our lives than events of just yesterday, but not necessarily.
Then again, not all Italians make full use of these potentialities; Tuscans, Romans and other living in Central Italy are most aware of them. Northern Italians tend to use almost exclusively passato prossimo, even for historical facts, while in parts of Southern Italy passato remoto is widely used as the unique past tense.