After doing some experimenting, I realize the description in the documentation is misleading. It makes it sound like each individual channel is added together to determine the Total, but from what I have found the Total is more like a Max Threshold per channel.
Here are some test renders:



At a Total of 2 you would expect there to be no bounces left over for Specular and Transmission because Diffuse would have already taken 2 bounces, rather it appears Arnold is giving 2 bounce per channel.

In the comparison above, the Total Ray Depth is the same (Total-Diffuse-Specular-Transmission) but you would expect when the Diffuse and Transmission are set to 1, that would allow more bounces in the Specular, but as you can see in the reflections the bounces remain at 4 for both renders.
As far as I understand everyones' explanation was wrong and Total Ray Depth is actually a per channel max, not a sum total of the channels.
Which makes sense if you want to globally dial the max per channel down during preview rendering as a threshold, and then remove the threshold during final renders.
What Arnold is NOT doing, is counting ray bounces between channels, therefore it doesn't have to prioritize between Specular and Diffuse for example when it "runs out of bounces" as the description "Diffuse + Transmission + Specular <= Total" would suggest.
Ultimately I think the word "Total", and the description in the documentation are misleading. I would suggest using the word "Max" instead, and describe it as a threshold that limits each channel to the Max number of bounces.
If anybody thinks I have it wrong please let me know.