tellner
Senior Master
I think the point isn't the variety of influences. It's being able to sythesize them strategically. Yes, you can study two (or more) fighting systems, but if you aren't able to integrate them into a single, systematic play of strategy and tactics, you wind up trapped like the fox in Aesop's fable, who has a hundred tricks but can't figure out when to use which one. You can learn a lot of things from different places, but I suspect you'll be doing yourself more harm than good if you don't find a way to synthesize them into a single consistent `operating system'. If you can, however, then you may have made real progress.
It's a big if, which is why it's probably better, for a lot of people anyway, to stick to one, already-worked-out synthesis of combat techs...
There's the heart of the matter, cleanly dissected and circulating gore.

You can't just memorize. You have to learn how to use it. And you have to learn how to integrate new knowledge. Or as Massad Ayoob puts it "It ain't simple and you ain't stupid."
That's really where JKD can shine and where it can fall flat on its prat. It can be a style like any other martial arts style - funny looking Wing Chun with very specific bits of boxing, fencing, a little bit of Judo and so on. "We do it exactly the way Bruce did from October 22 to March 4." Or it can provide you with a technical base and the tools to see what's significant about what other people do, extract the principles, training methods and cool tricks and integrate them into your own practice harmoniously.
The question of when you are ready to do that varies tremendously from person to person. It's part training, part natural inclination and part having a teacher who can show you how. Some never will. Some get it but need to wait until they have the technical skills and fighting ability to match their understanding. Others start out doing more than one thing. It's not an issue, just the most natural thing in the world. In Japan there are plenty of people who started Judo and Kendo in school and continue with them.
That's another point, come to think of it. If you're doing two things that are radically different, like wrestling and fencing, they won't interfere. If you're doing two things that are very close like Goju Ryu and Uechi Ryu the only problem is going to be minor confusions about specific techniques. But why bother? It's where things are somewhat different but close enough that there's overlap.
In Guru Plinck's Silat classes you can always tell the people who have a martial arts background from the ones who don't. The ones who don't have nothing to unlearn and tend to make faster progress for quite a while. But once they've been silatized the ones who came from something else find their old stuff coming out as needed even if it looks a little funny.
Children who grow up hearing only one language tend to develop speech a bit faster. Those who are in bi- or multi-lingual environments take longer to get started, but once they do they pick up other languages easily. Something similar is probably at work albeit in different parts of the brain.