I have a 15amp breaker using 12 romex going to two outlets in the downstairs bathroom & then to a outlet in bathroom upstairs. I have a roll of 14 romex. "Can I use 14 romex to add a outlet for the computer on the other side of the wall running off one of the bathroom outlets?"

Yes. If bathroom has a GFI, be sure to use screws not controlled by it.

Chuck

Well... if it's your house you can do as you wish...
Although the code directs bathrooms circuits to be dedicated to the bathrooms
So tapping for a receptacle in another room would not be keeping within the
Intent of the code language...
Bathrooms are dedicated to make available capacity for hair dryers which not only draw
A lot of power but also create a lot of electrical noise on the branch circuit which may have some negative effect on a computer...
If there was another circuit nearby, I would consider leaving the bathroom circuit dedicated.

No, for several reasons:

1) If the circuit serves two or more bathroom's receptacles it cab only be used for the bathroom receotcales.

2) The circuirt is required to 20 amps. #12 AWG supports 20 amps.

3) If you put a section of #14 AWG conductors the entire circuit must be reduced to 15 amps