Here in Germany bank accounts are pretty expensive. In the UK you are legally only allowed 1 current account afaik
The only account that is legally restricted that I can think of is the ISA which offers good tax incentives for saving but with a limit to how much can be contributed in a year, and I think you can only open with one provider in a given tax year.
If that's anything like the truth, absolutely nobody here knows about it. In an effort to get customers, banks will often give you "introductory" rates which are better than the market, but only for a fixed amount of time (6 months or a year). I know people who have accounts at basically every major bank.
Without the deal, there's no restrictions.