The exact circumstances are obviously very specific to your project ( e.g. development team, management style, process maturity, difficulty of the subject matter, etc.). In order to scope this a bit better so we can speak about it in anything but sweeping oversimplifications, I'm going to restate your question:
Under what circumstances, if any, can adding team members to a software development project that is running late result in a reduction in the actual ship date with a level of quality equal to that if the existing team were allow to work until completion?
There are a number of things that I think are necessary, but not sufficient, for this to occur (in no particular order):
One of the first things that should be discussed is whether the ship date can be slipped, whether features can be cut, and if some combinations of the two will allow you to satisfy release with your existing staff. Many times its a couple features that are really hogging the resources of the team that won't deliver value equal to the investment. So give your project's priorities a serious review before anything else.
If the outcome of the above paragraph isn't sufficient, then visit the list above. If you caught the schedule slip early, the addition of the right team members at the right time may save the release. Unfortunately, the closer you get to your expected ship date, the more things can go wrong with adding people. At one point, you'll cross the "point of no return" where no amount of change (other than shipping the current development branch) can save your release.
I could go on and on but I think I hit the major points. Outside of the project and in terms of your career, the company's future success, etc. one of the things that you should definitely do is figure out why you were late, if anything could have been done alert you earlier, and what measures you need to take to prevent it in the future. A late project usually occurs because you were either:
Hope that helps!
It only helps if you have a resource-driven project.
For instance, consider this:
You need to paint a large poster, say 4 by 6 meters. A poster that big, you can probably put two or three people in front of it, and have them paint in parallel. However, placing 20 people in front of it won't work. Additionally, you'll need skilled people, unless you want a crappy poster.
However, if your project is to stuff envelopes with ready-printed letters (like You MIGHT have won!) then the more people you add, the faster it goes. There is some overhead in doling out stacks of work, so you can't get benefits up to the point where you have one person pr. envelope, but you can get benefits from much more than just 2 or 3 people.
So if your project can easily be divided into small chunks, and if the team members can get up to speed quickly (like... instantaneously), then adding more people will make it go faster, up to a point.
Sadly, not many projects are like that in our world, which is why docgnome's tip about the Mythical Man-Month book is a really good advice.
Maybe if the following conditions apply:
I'll let you know the first time I see all of these at once.
According to the Mythical Man-Month, the main reason adding people to a late project makes it later is the O(n^2) communication overhead.
I've experienced one primary exception to this: if there's only one person on a project, it's almost always doomed. Adding a second one speeds it up almost every time. That's because communication isn't overhead in that case - it's a helpful opportunity to clarify your thoughts and make fewer stupid mistakes.
Also, as you obviously knew when you posted your question, the advice from the Mythical Man-Month only applies to late projects. If your project isn't already late, it is quite possible that adding people won't make it later. Assuming you do it properly, of course.
If the existing programmers are totally incompetent, then adding competent programmers may help.
I can imagine a situation where you had a very modular system, and the existing programmer(s) hadn't even started on a very isolated module. In that case, assigning just that portion of the project to a new programmer might help.
Basically the Mythical Man Month references are correct, except in contrived cases like the one I made up. Mr. Brooks did solid research to demonstrate that after a certain point, the networking and communication costs of adding new programmers to a project will outweigh any benefits you gain from their productivity.
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