Two things. 
Why are you re recording a sample from another computer? Can you not just simply transfer the WAVE / FLAC or whatever file with a USB stick? Or at least use "what you hear"  as a input and record it internally. 
 
This does sound like a clocking issue as you are using a chain of digital devices. 
The clock would have to be driving the first computer, the RME interface, the 02R96, the ADAT card and the recording DAW....This is a job for a master clock source for sure. 

This is not a clock issue... 100% guaranteed everything is clocking rock solid via the Big Ben.  There's a **** ton of other digital stuff I haven't even mentioned (Finalizer, Lexicon PXM-91, Yamaha SPX2000, DAC-ST, Roland M1000, etc) believe me I've experienced every possible clocking issue and there's nothing that is not properly in sync.  Also, transfering via USB stick sounds like a nightmare.  Not to mention my computers are housed in another room, not to mention it's a keyswitched sample I wouldn't even know where to begin to look for the actual waveforms.  It takes two seconds to arm a track, hit record, and play the sample.  Lastly I regret posting that sample, that was just an example of the digital distortion I was getting, not really the true problem I was having, and I haven't heard it once since I changed from 16-bit to 24-bit...

Please don't take this the wrong way, but you should go and read up on gain staging.
 
Getting your levels set correctly all the way through the chain is the easiest & most effective way to avoid these sort of problems

Of course I don't take it the wrong way, you guys are a huge help and resource to me.  I do think I am unable to explain the way I work well enough for you to understand exactly what I was trying to fix.  If you want to keep helping I am happy to elaborate but my issue is [Solved!]

Based on what you've said, what solved the problem, and what I saw in the recording, I suspect there was some sort of "wrapping" of amplitude between peaks and troughs going on due to mismatched bit depths. I've seen some cheap audio interfaces do this when the input level exceeded the limit of the A/D converter. Instead of clipping it, the converter would "wrap" the sample value around to the other side of the waveform. Some sort of Two's Complement programming error.