Today they got in touch, to say that a customer had told them he had filled in the contact form, but no email had come to them as a result.
I checked the form - dated last updated 2003 - and found that when submitted, it attempted to send email by calling a file that doesn't exist. This file has, in all probability, not existed for at least 3 years, since I don't have a copy in the snapshot of the site I took when I took it over. It may, indeed, never have existed at all.
The moral of this story is, if anyone ever builds you any kind of form, don't just look at it and assume it works. Fill it in!