Did you tried AT&F ..?
Try checking the module's registration status i.e AT+CREG? command. Make sure you module doesn't loose registration to network after receiving 1-2 messages. Also keep checking signal quality using AT+CSQ command.
If the module remains registered and signal quality is also good, then make sure that your SIM is not already full with messages. Try AT+CMGD="ALL" to delete all existing messages and empty the SIM's memory.
I still the problem persists, then i would say module's faulty. Order a new one rather then wasting more time on it.
Regrads,
Salman Liaquat.