If you're after a name from a mobile number then no, it cannot be done without breaking the Data Protection Act which would get you in a whole world of trouble. If someone working for the relevant network did a look-up for you, they'd face prosecution too.
BT used to have something like a 'phonebook on a disk', not sure if they still do it but that's for landlines only.
Evevn the directory enquiry companies wouldn't be able to do it (I used to work for 118888) - they search on the name, not the number - but in any case, they list landlines only.