I can't think of many examples when this is the case. For newcomers I'd rather say the opposite - adding one mod at a time is the safest move as it's easier to trouble shoot that way.
Anyway, one thing to keep in mind is that you should wait for a respawn when installing/updating mods that alter the leveled lists or add new ones, as for example MMM. Install the mod, wait 4 days (if on default respawn) and you should be good to go.
I guess I should have elaborated on that. I meant that adding a mod to a new game after you are sure that the setup works is safest. Adding mods one at a time is definitely the way to go, especially to determine whether or not a certain configuration works. If you look at the ReadMes for certain mods, usually those with significant quests, they may recommend starting a new game with the mod. The same goes for overhauls that relevel NPCs. If you have Wrye Bash there are ways to update most things, but not everyone uses Wrye Bash. Another example is RBP, and I guess other mods that modify birthsigns and the like as well, which requires an update script if the user adds the mod to an existing game. The other issue is that existing games are Form ID sensitive. One of the issues that can spring up out of that is the floating arrows bug that Blade of Mercy mentions in his guide.
One at a time is good, but I think adding one at a time to a relatively fresh game is even better. Add a mod, which can be multiple plugins too, play it to test it out. Add the next mod, and do the same... Once you have a setup you are comfortable with, you can start a new game, where you do the one by one loading method, and play with all of your desired mods from the beginning, not missing any features. I do not see how these views are "opposite." I just seem to be on the wrong side of the fence everywhere now...