However, the clearing banks are now playing catch-up and Barclays, which last Monday opened its own private banking division, is the latest of the high street banks eager to tap into the large and lucrative market of HNWIs - high-net-worth individuals.
A report by Datamonitor - which counts HNWIs as those with liquid assets of pounds 250,000 or more - reveals that not only are the rich getting richer but their numbers are increasing: 1.6 million by the end of 1994, who between them owned assets approaching pounds 200bn. The increase is partly due to the fact that, as a population, we are ageing and partly a result of the growing entrepreneurial class spawned by the Thatcherite economic policies of the 1980s.
The report concluded that even those individuals at the bottom of the high-net-worth scale have financial needs that differ noticeably from the bulk of the retail market. Very simply, HNWIs - accustomed to flying first class and being shown the best tables in the best restaurants - also expect that little bit extra from their banks.
They are also willing to pay handsomely for good service, and Coutts only recently signalled a return to its roots by abandoning its attempts to move down-market and concentrating instead on its core of HNWIs.
Lloyds was the first of the high street banks to open a separate private banking arm back in 1989. David Maguire, head of marketing, explains the rationale behind the move. "Many of the retail banking outlets are simply not specialised enough to give detailed planning advice to very wealthy clients. Virtually all the clearing banks are tied to their own life companies, for example, and the branch network is geared towards packaged products, which in many cases aren't suitable. Many of our wealthy clients require a much broader range of services - derivatives dealing, for example, or currency accounts."
Most of the traditional private banks - and all of the private banking arms of the clearing banks - are eager to stress that what they are offering is definitely not merely "red carpet" retail banking - leather chequebook holders and regular drinks with the manager.
"We are not interested in attracting the sort of customer whose main reason for banking with us is to get hold of an unusual charge card which they can wave about ostentatiously at supermarket checkouts," said a director of one of the older private banks.
It is a sentiment shared by most of the industry, for the real money to be made in private banking comes not from running current accounts for wealthy clients, but through management of the clients' total assets. Barclays calls it the "asset base", citing what it hopes will be typical scenario of a client who might have a trust fund in Jersey, a cash portfolio in Switzerland and an equity portfolio, all managed out of the plush offices of Barclays Private Banking in London.
Barclays says there is no entry threshold for its private banking service but, eager to avoid confusion with its Premier Service - already in operation for its wealthier high street clients - lets drop the fact that their average private banking client has over pounds lm under management. This, then, is a service for the seriously rich.
Such high entry limits are unusual however - certainly with the older private banks. Anthony Townsend, a director of Rea Brothers, founded in 1917, explained that nowadays the majority of people with large disposable incomes have made their money rather than inherited it.
"Our fee structure is not aimed at those who just want a cheque book but we do recognise that not everyone starts off rich and what we're really looking for is the prospect of a long-term relationship." he says.
"We would happily look at a portfolio of pounds 50,000 if the client had a growing business and we thought that client could make best use of all of our services."
Such services might include advice on asset or cash management or help from the corporate finance division.
The Private Bank Company is one of the newer independent private banks. Established in 1989 by the Greek Latsis family, its start-up capital was the highest of any UK bank - pounds 100m. In addition to the usual services any retail client would expect, the bank provides treasury, foreign exchange and interest rate management services. Property management and inheritance planning are also in demand and, again, the corporate advisory division provides an important service for those clients who run their own businesses.
"Close relationships are the key to any financial arrangements - whether you're banking in the high street or with a private bank." says Gerrard Gardner, executive director of the bank. "The difference is that private banks like ourselves are prepared to spend money resourcing that relationship.
"Wealthy people do not expect to phone their bank with an inquiry only to be asked what their account number is by some anonymous teller at the other end who proceeds to pass the call around the office."
Rea Brothers' Mr Townsend summed up the philosophy behind being a private banker: "It is rather like being a good private doctor." he said. "If you don't know the patients' overall health then you are unlikely to prescribe the right medicine." For those fortunate enough to count themselves NHWIs, there is no shortage of doctors queuing up to prescribe it.