Combine multiple major fellowships? I have received both the NSF and another major fellowship (non-federal). Both stipends cover more than my departmental funding; do I get the sum of the stipends?
First of all - congratulations! 

To answer the question, the NSF provides specific advice on this question: 


  May I accept or be paid from a private Foundation Fellowship?
  
  Supplementation to a Fellowship while on Tenure is at the discretion
  of the GRFP Institution. Fellows should check with their COs.


In practice, I've found that you can at times get funding to combine with the NSF, but it is often intentionally limited by your local institution/department. For instance sometimes you can get summer funding or a 0.25 FTE assignment even while receiving the NSF, but you may get administrative pushback, and they are loath (or simply will not) allow more than that. 

You will generally have a higher likelihood of getting them to agree if you can take the funding at different times. The NSF GRFP for instance is for 5 years, but only 3 years includes funding. If you have a fellowship that provides 1-2 years of funding, you can take that first and then stay "on reserve" with the NSF - and most institutions will be somewhat unlikely to object to that. One institution I talked to offered a 1 year fellowship, and on asking them they replied that I could take the 1 year and then take the NSF funding in other years, but that I would not be permitted to take them both in the first year.

But ultimately, the NSF says that is up to your local institution, so you'll have to check with your department to see what they say. 

For completeness, note that the NSF explicitly cannot be combined with other fellowships provided by the US government (they mention "DOE, EPA, NIH, NOAA, USDA, DHS, DOT, NASA, etc."), but as the OP mentioned non-Federal this should not apply to them.