Diana, I think Neil has hit the nail on the head here, you haven't been refused insurance by an insurance company or underwriter (which is usually done because of bad driving/claims record or non disclosure of previous convictions etc.) but merely not met the random requirements of a broker, who rightly or wrongly thinks you don't meet their 'specialist' requirements.
Maybe speak to the owner/someone at the top and try to pin down what exactly is the problem, even if you get a ridiculous quote that makes you feel better than refusal.
They work in mysterious ways sometimes, one broker really really wanted to insure Rugged Robin so he could use it in their advertizing as an example of how they could insure any vehicle, but he was unable to come anywhere hear the quote I already had from another broker, so I didn't go with him.