Your question covers two vaiables: exposure EI and development time. If you expose for EI 1600 and develop normally (N) you'll get a thin negative (all black shadows and muddy mid tones.
Yes, it will be thin. That thinness and the muddy mid-tone issue is correctable with a harder paper grade; film development changes (like a push) and paper grade changes (like moving to grade 4 or 5) make essentially equal changes to the final result. In theory a 2-stop push is equal to a 2-grade paper change.
If you overdevelop two stops (N+2) you'll get normal shadow detail and blown out highlights ( the sky f.ex.). This can be effective if your subject contrast range is very low (a landscape on a very dull day f.ex.). Now figure out what would happen if you do both!
When the camera exposure is reduced from box speed the normal shadow detail on the negative is reduced, that doesn't necessarily affect the print immediately but it is a fact. Film ratings have some safety factor built in and on top of that what we expect in the shadows might be low. That safety factor helps, as do low expectations for the shadows, but with a camera exposure 2-stops under box speed normally available print detail is probably being lost. At that point we have essentially lost the ability to dodge for more detail.
But this reasoning is based on the results from printing on paper. I don't know anything about PS and LR.
Peter
The details aren't important here on APUG, but practical proof is handy.
Thousands of rolls of film from disposable cameras are processed by labs every day.
These cameras typically use 400 speed film (Ilford even sells their own with HP5 installed). These cameras have no exposure adjustments, shutter is normally about 1/100th, with f/11 or so for the aperture.
The EI is the exposure variable for each shot and it simply "floats". Some shots may end up at EI 12 (and will have a lot more than normal shadow detail that you may or may not use) others may end up at EI 1600 (and have no reserve shadow detail).
Regardless of how each frame is shot the film is developed normally and the contrast and placement are adjusted for printing digitally.
What we get back from the lab may not be perfect, but the lab's first try should IMO be called proofs; still and yet these proofs typically turn out just fine, or at least fairly close in most cases.
