There can be many reasons why you have been refused a debit card. Many banks have their own scoring system and although they use the services of companies such as Experian and Equifax they will have their own internal points system.
I would initially speak with the bank. If you arent satisfied with their answer then contacting the 2 companies I have mentioned and request your credit report. You may find you are associated with someone elses credit history.
Get to the natwest!! Tell them you do not have any other bank accounts and you will get a bank account without overdraft (which is good really) and a solo debit card!! Good luck.