Don - Lloyd's is not just one underwriter, it is hundreds all doing business, as syndicates, in their own name but under the umbrella of Lloyd's. If one underwriter has been bad paying out you cannot tar the entire enterprise with the same brush!
There are also conglomerates of companies that operate parallel to Lloyd's. A broker works for the customer but is paid by the underwriter, strange but true, a good broker will be responsible for resolving any disputes about pay-outs. Single biggest cause for an underwriter rejecting a claim is non disclosure of essential information by the insured, again, something a broker should ensure doesn't happen.