The owners of the house, Stephen Ponder and Caroline Culley, have both buildings and contents insurance. They are also covered against accidental damage.
But a legal quirk means the only way they can get one of their insurance policies to meet the pounds 200 cost of repairing their neighbour's car is to persuade him to sue them.
Their neighbour would not be entitled to compensation under all circumstances. If the slate had come down during a storm the claim would fail because the neighbours could not pursue a claim for negligence against the home owners.
The couple, from Claverley, near Wolverhampton, are being told by their buildings insurer, Legal & General, that there is a legal difference between their duties as occupiers of their home and their ownership of it.
Because of this separation, under which Legal & General classes them as owners of the property, the company is unable to pay for the damage to the neighbour's car.
Liability for this rests with the home's occupiers - also the couple - who are covered under a separate contents insurance policy, taken out with another firm, Frizzell.
Frizzell has told the couple that in order for it to consider the claim the neighbour should claim against them for negligence. It would then look at the evidence to see whether the claim should be met or opposed.
Ms Culley said: 'We know that the damage in this case was slight and the cost only comes to a few hundred pounds. But what would happen if instead of a slate it had been the chimney collapsing? This kind of thing should not be allowed to happen.'
Mr Ponder said: 'This is very frustrating. First, a company which commonsense suggests should be meeting a claim of this sort denies responsibility. Then we are told that we should get our neighbour to sue us, presumably all in the interest of good relations between people living next door to each other. Either that or claim on his car insurance, which would really do wonders for our relationship.
'I am left wondering why we are paying so much for our cover when companies are able to make these distinctions.'
An L&G spokesman said: 'The law places a duty of care, and therefore one of liability, on the occupiers of a building - presumably on the basis that they are more likely to be aware of its condition.
'It has become standard practice for insurance companies to make the distinction in respect of their policies. Accordingly, owners' liability cover is attached to buildings insurance and occupiers' liability cover is attached to contents policies.'
The spokesman added that, following a recent ruling by the Insurance Ombudsman, a couple can claim on their buildings insurance for accidental damage to a neighbour's property if they do not have contents cover.
Clive Brewer, claims manager at Frizzell, agreed that when his firm was the contents insurer it would normally be responsible for meeting such a claim. But he added: 'For a claim to be met, the third party, in this case the owner of the car, must serve a claim on the insured party. When we examine a case we look at whether the home's occupiers were negligent or failed to exercise their duty of care.
'If the couple contact us we will look at the case. These cases do not normally have to go to court. But the question of liability is important. This is not to say that we attach any blame to the insured parties themselves.'