Besides three elixir sessions per day, some banks offers instant transfer (about 15 minutes) for detail clients, up to 5000 zł.
Anyway Elixir (and EuroElixir) is only one of the couple of settlement routes. For big transfers between banks there is Sorbnet, and soon Sorbnet2.
Besides that many banks are SWIFT and TARGET participants. Obviously there are also transfers through SEPA.
PS. this instant transfer settlement route is in fact private service by Blue Media (?) which technically uses their own accounts in banks in order to take advantage of instant internal transfers.
Do larger monetary amounts sometimes postpone transferring of funds?
As far as I know it largerly depends on your personal agreement with a bank (limits of transfer are part of this).