Oversampling and Decimation

Hello all, 
I'd like to know what is the limitation of oversampling in terms of using the over sampled value. 
I am using DSPIC30F5011, I want to achieve 16-bit resolution from 12-bit. 
I know that I have to sum 4^n (where n is number of extra bit resolution). So for 13-bit resolution from 12-bit I should sum 4 12-bit values and right shift by n ;
- Therefore in my case 4^4 = 256, lets say that I am sampling a 5Vref pin, this is 4095 in terms of 12-bit. ( 5V/4095 = 1.22mV ) in my case.
- So, sum of 256 values (@ 5V ) of 12-bit is 1048320, right shift by 4, is equal to 65520 ( 1111 1111 1111 0000 ) { 2^16 = 65520 }
- My question is after I over sample will I have any problems storing or using this result in my 12-bit ADC PIC ? Any data size restriction? 
 

Thank you Dario for your quick response. 
I will go on with my plan. 

The chip itself produces this noise intrinsically, so oversampling and decimation works just perfectly. I use it widely and get the 16 bit resolution easily (do not mix with accuracy which stays intact).