One of the good things about software is that people seem to want it, and want it quickly. It's usual for organizations to ask teams to speed up production of software and from time to time the organization seeks to help in the way that really shows its commitment - by spending money to add more people to the team.
Now there's many an argument that's been had about the true benefit of adding people to a software team. It's clear to me that you don't get a linear benefit, doubling a team won't double its productivity because the communication and coordination costs kick in and blunt the increase. My utterly unscientific rule of thumb is that any increase in productivity is likely to be proportional to the square root of the increase in people, so doubling a team will get you roughly a 50% increase in productivity. But in practice this will vary a great deal depending on individual factors. There are some people I know who are likely to double the productivity of even a sizable team just on their own, and we have run into people who will reduce a team's productivity.
But the issue I want to highlight here is that of the ramp-up. You have a small team that's working well, but you want more software and you are prepared to spend the money to get it. You're happy to pay quadruple, even sextuple to double your rate of progress. An important, yet not well understood factor is the rate at which you can safely add people to a team.
More than once, I've come across projects who added too many people too quickly. This manifests itself in a breakdown of the cohesion of the code base itself. Duplication runs rampant, several friends of mine know about the project that had three object-relational mapping frameworks within a single application. This breakdown occurs because the new people don't understand how the code base currently works, so they do something at odds with it, like adding a competing framework. If there's too many new people around, the team leadership can't keep track of it all and the code base sprouts problems. These problems then reinforce each other because nobody can find a consistent way to do things, the Broken Windows syndrome kicks in and you get a positive feedback loop. (And positive feedback loops are usually a bad thing.)
On top of this an overly rapid ramp up leads to a break down of the human communication mechanisms. It takes time for people to get used to working with each other and a rapid ramp up can stop a large team from forming the relationships it needs to succeed.
So how much ramp up can you safely do? It's difficult to give any concrete advice here, because any experienced project manager I ask always rightly points out that there are many variables that have to be taken into account. I pressed Joe Zenevitch, one of my most trusted PM sources, and he indicated that he would never want to more than double a team at once. Even doubling would be a risky call, with the risk increasing if the exisiting team was already a dozen or more people, or had a signficant amount of juniors.
If you do a significant increase in size you shouldn't add yet more people until the new folks have settled into the team. It will take a few weeks for this to happen. Developers need to get to know the code-base and the domain, BAs need to be familiar with the domain experts, everyone needs to get to know each other. At ThoughtWorks we expect people to come up to speed quickly, after all we hire bright, high motivated people who are quick learners. But even so it still take a week or two. For most teams you should allow a good bit longer.
When you add people to a team, you don't get an immediate increase in capability. It takes time for people to become productive on a new project. Worse still existing staff have to spend time helping them get up to speed, so your velocity may well drop at first. Joe Z's observation for ThoughtWorks teams is that there will be no net effect for the first couple of weeks as the new people and hit for existing people cancel out.  Most ThoughtWorkers like to point out that pair programming is a big enabler to on-boarding people more rapidly. Pat Kua also has some good advice on how to bring new people onto a team effectively.
Another thing to watch is to not ramp-up too early in the project. One of the firmest bits of advice I've heard from people who do large projects (anything over 50 people) is that the project should start small, maybe with just a dozen or so developers. They should figure out the key design elements and interactions of the system by building early parts of it. Only once that design has settled down should you think of increasing the team size to its full size. As part of that settling down put time into removing any design elements that you don't think should be copied. People will naturally copy stuff that's already there, so you should ensure that what's there is all going to make a good platform for further development. This is a time to err on the side of excessive attention to code-cleanliness.
Finally, when thinking of ramping up, think very carefully about whether it's worth it. I've rarely come across a large team where there isn't a feeling that the team could be significantly cut without reducing its productivity. As I once said "scaling agile is the last thing you want to do".
My colleague Francisco Trindade talks about a good experience he used for bringing on a few developers at once over a couple of weeks.
1: This couple-of-weeks rule is for ThoughtWorkers, so we would expect it would take longer for people who don't match our hiring standards.