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Pixel depth will be dictated by the file format and application. E.g. TIFF is largely a 16bpp format (the latest version supports 32bpp) but, for example, until recently GIMP couldn't save TIFF files with greater than 8bpp, so it truncated on save. Edit, that's X bpp per colour channel .. and there may be an alpha channel, so it can get very confusing.
If you have three 16bpp images in the same format and "sum" them, then whatever algorithm the application is applying in it's "sum" function (which could be weird and wonderful), the final image is still going to be 16bpp ... unless you can also change format to a deeper bpp format and assuming your application holds that data all the way to save.
Keep in mind, however, that those extra bits are at the least significant end, so you're losing precision, not brightness or saturation when truncating from, say, 16bpp to 8bpp.
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