It's probably useful to distinguish between playing more than one character sequentially and more than one character at once. I'm assuming from your question that you mean at once.
I'd say it's acceptable any time your group is comfortable with it. Some games explicitly call for this: Apocalypse World gives players the ability to play multiple PCs as part of advancement, which I think is very clever: new players won't be able to do it, but once you've played the game long enough to really get a feel for it, you have the option. Early D&D also more or less assumed you'd have henchmen and so forth, which is not exactly the same thing, but it might have the seeds of useful ideas.
What if, rather than making the extra characters full PCs, you used some sort of companion rules? Strip down the characters to their essentials, explicitly assume that they won't get as much roleplaying attention, and so forth.
Two specific ideas in that regard: one, reserve the right as the GM to control companions if you ever want to. I would use this as a mechanism for the companion to say no to orders, mostly, and I'd use it fairly rarely -- think of it as a morale thing. "Dude, you ordered Bob to go down there and he never came back."
Two, make them pool characters, so that different people run them at different sessions. That will have the specific effect of minimizing immersion in the companion characters, so make sure you want to do that as opposed to increasing identification.
I think no matter what you do, this is going to affect your immersion a bit, simply because you'll have to make decisions about more than one character. But focusing on the main PC should help some.