Please bear in mind that your bank (or the payee Bank, in general) must be the same as of the drawer's, for the Bank to be obliged to pay you the money immediately (either if you deposit the cheque in your checking account and have the money available to use it whenever you wish, or if you cash/withdraw it at the teller).
If your checking account is in a different Bank from the drawer's, the depositing rules between different Banks apply, meaning you will have the funds available only after the clearance of the cheque through the Clearing System.
Of course that, as Just Play says, if you have a good relationship with your Bank (assuming it's not the same as the drawer's) they may make the funds available immediately for you (in total, or partially), but they are not obliged to.
I have never heard of partial payments from deposited cheques either, although I know that any Bank may give funds in advance to any customer they wish to, something different.
But please excuse me,
this is somehow incoherent, because you are saying that your Bank makes advanced/partial payments from deposited cheques for you too
Anyway, for what you said about your relationship with your Bank, they are just giving you a credit in advance for you to use, because they trust you will be able to cover it, if by some reason the cheque you have deposited bounces.
That was probably what happened with Mousey (and his/her? Bank was certainly able to clear the cheque in few days, but they used the funds for some time while they were releasing/dropping it partially into Mousey's account).