In general it will RSS (root-sum-square -- add up the sum of the squares of the noise/distortion terms, then take the square root) but for harmonic and intermodulation distortion may be closer to additive (phase differences cause amplitudes to vary). Of course you need to account for the gain through the chain. We do this all the time for RF systems; there is a trade among gain, noise, distortion, and power that can get gnarly at the system level with lots of elements cascaded. I used to have a spreadsheet for basic what-if trades but used a simulator and then measured for better accuracy.

FWIW, the rule of thumb is that a source 10 dB lower contributes "negligibly". I have a chart, somewhere, showing the relative contributions to the net output for various distortion/noise levels.