I think it really depends on:
(1) which sports you want to do;
(2) how similar they are to each other;
(3) how much experience you have in training each sport;
(4) how competitive you want to be; and
(5) your dog's athletic ability, mental acuity, and willingness to work.
Some sports -- lure coursing, dock diving, barn hunt, etc. -- require little or no training. Either your dog wants to do them or he doesn't. If he does, it's pretty much plug-and-play. If he doesn't, you probably could train them, but I can't really see why you'd bother. These are just-for-fun activities; I don't think anyone takes them seriously as a measure of the dog's ability or the trainer's skill.
Other sports require much more training. Agility, IPO, upper-level obedience -- all of those entail months to years of work before you're ever ready to set foot in a ring, and more months or years if you want to reach the top.
The next factor to consider is how similar the sports are to one another. This has pros and cons. On the up side, the more closely one sport resembles another, the less new training you have to do to make the crossover. If you're already training in IPO, then the requirements for AKC Novice obedience really don't require adding on a whole lot of new stuff. If you've trained Utility scent articles, then introductory nosework (to about ORT readiness) is pretty simple.
The downside is that the more closely one sport resembles another, the more likely it is that you and your dog may get confused on the finicky small things that can cost you points in the ring. I managed to lose 20 points(!!) on an AKC Rally Advanced run because I executed the exercises in the way that would have been correct for World Cynosport Rally but was not correct in AKC Rally. Same sport, different rules, worst score ever.
Similarly, I notice that whenever I've been doing a lot of obedience-style heelwork with Pongu (long flowing stretches), his Rally-style heeling (choppy short spurts with lots of starts and stops) tends to suffer, and vice versa. Doesn't matter for Qs, does matter for scores.
Then there's trainer experience and dog experience. As I write this today, I'm pretty comfortable training all the Rally exercises and most of the obedience exercises. I'm a total novice to agility, as is my dog. It's therefore harder for me to train the agility exercises because I really don't have a clear picture yet of what the final finished forms are supposed to look like or how the nuances in training them might influence the outcomes. This means it takes more time and effort for me to reach X level in agility proficiency than it does for me to reach the same level of fluency in other sports that I'm more familiar with.
However, Pongu has a really strong foundational background from other sports and I am pretty good at clicker shaping, so we've pretty much condensed six months of agility foundational work into six weeks. There are some shortcuts we can take after training in other sports that genuine newbie teams can't do.
Then there's the level of competition you want to reach. If you just want to play around in a new venue and have fun, it's not that hard to hopscotch among them. If you want to really master a field, though, it is harder to train simultaneously in multiple sports because you're dividing your efforts and there's only so much time in the day, and only so much mental and physical energy that you and your dog can devote to perfecting the fine points in each sport.
It's also obviously harder to trial in everything at once, since you'll run into scheduling conflicts and at some point most people run out of money for seminars, practice equipment, specialized classes, entry fees, gas, etc. Especially if you're trialing multiple dogs in multiple sports, this can add up FAST.
And then, finally, it varies based on your dog. Some dogs can handle more than others (there's a reason those go-go-gooo!! Border Collies are so popular among cross-sport competitors: they're smart, they work hard, and they never seem to get tired). Some seem to find it easier to keep all the different "jobs" straight in their heads (although, of course, trainer skill makes a big difference there too). Some can't handle that intensity, though, and will burn out.
Personally I think there's much to be gained from training in different sports. You learn different things and get to try out different approaches, and all those skills add to your problem-solving toolbox.
But I also tend to concentrate my trialing efforts in one thing at a time. Right now we're phasing out of Rally/obedience (will probably wrap that up at the end of this year) and shifting more into agility (will probably start trialing in that sport sometime in early 2015). I'm not into dabbling in a lot of things at once; my preference is to concentrate on one thing, achieve whatever level of proficiency I can in it, and then move on to the next.
ARCHMX TDCH Pongu the Insane, CD-C, RE, RL1X6, RL2X5, RL3X2 (GSD mix, b. Apr 2010)
Crookytail the Tigerwuff, RL1, ITD (Akita mix, b. Jan 2011)