The problem is not with the cost of these changes - which is running into hundreds of millions of pounds - but with the number of people and the amount of time available to make the changes.
Kit Farrow, director of the London Investment Bankers Association (Liba), said: "It's a question of whether we're attempting to achieve more than collectively City institutions can cope with."
Crest, the new electronic share system, is continually dogged with problems and the introduction of new shares to the system has been delayed.
Problems with Crest have had a knock-on effect on the Bank of England's plans to upgrade the Central Gilts Office electronic settlements system which will use the same software as the Crest system.
This means that plans to trade the interest payments on gilts separately from the principal - known as "strippable" government bonds - have been pushed back further into 1997.
On top of these immediate changes, City banks are preparing their computer systems for the introduction of the euro, the single European currency, in 1999. Regardless of whether Britain takes part in the unification, banks must be ready to trade the currency.
They must also deal with the problem facing companies world-wide - the fact that computer systems were not designed to cope with dates after 1999. When 2000 begins, many firms will find that their computers believe it is 1900 again as they only read the last two digits of dates.
"We're turning our attention increasingly to the load on systems changes," Mr Farrow said. Liba was looking at this issue more vigorously than in the past and was talking to its members about the problems technological changes could be posing.
"We're very anxious not to jump to conclusions," said Mr Farrow. "But we may find that there are real questions to try to explore with members."