They may not realise that they can make a claim, or even if they do they are unsure how to go about it or how to get the best advice.
The problem has been addressed in a number of ways. The Law Society established a specialist panel for personal injury lawyers, and launched an 'accident line' in the summer, a free telephone line offering referral to a local specialist solicitor. Individual firms have not been slow in setting up a similar service.
It would seem that there is plenty of work to go round. More than three million people were injured in accidents in Britain in 1992; between 20 to 30 per cent of them claimed compensation.
The Law Society's accident line received 17,000 calls in its first three months, of which 11,000 were referred to solicitors. The extent of the line's success came as a surprise to the society, which had expected perhaps 10,000 calls in the first year.
So far, 1,200 solicitors' offices are members of the Law Society's scheme.
Two examples of firms that have set up their own versions are Thompsons (two associated firms, Robin Thompson & Partners and Brian Thompson & Parters), who are arguably the leading personal injury solicitors in the country, and White & Bowker, a 16-partner Winchester firm.
In June, Thompsons pipped the Law Society to the post by launching a compensation line, initially as a pilot exercise in Birmingham and Manchester. Stuart Henderson, a partner with Robin Thompson, explains that the line was set up from a mixture of altruistic and business motives.
'We feel we have considerable expertise,' he says. 'We believe there are a lot of lawyers without the same expertise who because of slick marketing campaigns or advertising are getting the work, when we are not.'
Easing the path to claiming compensation is therefore in the interest of the injured party as well as that of Thompsons, Mr Henderson says. And many of the people injured in accidents who could claim compensation don't because they are afraid of the costs that may be involved.
'They are concerned that once they are in the legal system it will become very expensive, so they take no action at all,' he says. 'We have tried to make it easier for these people to get access to a lawyer.'
The initial contact is free, and the arrangements for legal costs thereafter are flexible, says Mr Henderson, with a clear idea of the likely amount involved given to the client at an early stage.
The line has so far taken 'hundreds' of calls. The majority have been referred to other routes or advised to take no action. For example, a caller relating a tale of possible medical negligence may be advised to go back to the hospital for an explanation first, or approach the national health ombudsman.
The two Thompsons have between them 17 offices in England, Wales and Scotland. The firms have long-established roots in the trades union movement and act only for plaintiffs, never defendants or employers.
They have acted in a number of high profile cases, notably that of Christine Leung who won a record pounds 3.4m compensation following a car accident.
Last year, they won pounds 109m for clients for work-related injuries alone.
Eighty of the two firms' lawyers are on the personal injury panel, and the firm can also be reached via the Law Society accident line. Thompsons' own compensation line is about to be launched nationally, with each office marketing the scheme at local level.
One problem facing personal injury lawyers making any attempt at marketing that particular area of expertise is the accusation of ambulance chasing.
The charge irritates Mr Henderson. 'It's a terrible situation to be in,' he says. 'Lawyers are on the one hand accused of being stuffy and inaccessible, and then when they try to change that they are accused of becoming like American lawyers. Our view is that we are not,' he says. 'What happens in the United States is that you get people who are not properly qualified hanging around road accidents and hospitals, handing out their cards and literally chasing ambulances. What we are doing is making access to law easier.'
White & Bowker exists as the result of a merger four years ago of two of Winchester's oldest law firms. It has an established private client practice, but is increasingly focusing on commercial work.
Its accident freephone service began in April. It operates very much like Thompsons' and the Law Society's: a free telephone call, a free initial interview with an accredited personal injury expert (White & Bowker has three lawyers on the PI panel) and a clear idea of probable costs.
It is not the only Hampshire firm to offer such a service, says its director of marketing, Martin Davies. But, he says: 'It is one way of being proactive in the legal market, getting people to talk to you. The field of personal injury is a big market of people who don't seem to talk to anyone.'
Publicity for the scheme has had to be constant, Mr Davies says. 'You need to keep putting the name and number in people's minds, they forget very quickly.' Initial advertising on local radio and press was followed up with a Yellow Pages entry and a deal with a Winchester hospital, whereby the firm's name and details of the service are printed on the back of out-patient appointment cards.
Martin Davies says that around pounds 40,000 of business has come in as a result of the line. 'It is business we wouldn't otherwise have had and is not to be sneezed at,' he says.
The line has been sufficiently successful for the firm to consider different areas into which to extend the idea: a price-sensitive area or matrimonial work. 'At the very least,' says Martin Davies, 'a service such as the accident freephone creates name awareness and a feel-good factor, and you never know when it may come home to roost.'