At the time I write this, there are a number of wise words already written here, so I'll add just incremental thoughts. (Much of what I might say was already said by Matt Emerton.)
EGA and SGA are dangerous, because they are so powerful, and thus so tempting. It is easy to be mesmerized.
Their writing is roughly synonymous with the founding of modern algebraic geometry as a field. But I think their presence has driven people away from algebraic geometry, because they give a misleading impression of the flavor of the subject. (And even writing "flavor" in the singular is silly: there are now many vastly different cuisines.) But this is less true than a generation ago.
Here are some well-known negative consequences. Relatively few successful practicing researchers hold these views, but I have heard them expressed more than once by younger people.
There is a common feeling that there is an overwhelming amount one has to know just to understand the literature. (To be clear: one has to know a lot, and there is even something that can be reasonably called a "canon" that will apply to many people. But I think the fundamentals of the field are more broad and shallow than narrow and deep. It is also true that much of the literature is not written in a reader-friendly way.)
Those who can quote chapter and verse of EGA are the best suited to doing algebraic geometry. (To be clear: some of the best can do this. But this isn't a cause of them being able to do the kind of work they do; it is an effect.)
If you can’t use “non-Noetherian rings” in a sentence, you can’t do algebraic geometry.
(This is just as silly as saying that if you don’t know the analytic proofs underlying classical Hodge theory, then you can’t call yourself an algebraic geometer. It depends on what you are working on.)
Your goal is to (eventually) prove theorems. You want to get to “the front” as quickly as possible. You want to be able to do exercises, then answer questions, then ask questions, then do something new. You may think that you need to know everything in order to move forward, but this is not true. Learn what you need, do some reading for fun, and do no more.
Don't forget: EGA and SGA were written at the dawn of a new age.
The rules were being written, and these were never intended to be final drafts; witness the constant revisions to EGA, as the authors keep going back to improve what came before. These ideas have been digested ever since. Just because it is in EGA or SGA doens’t mean it is important. Just because it is in EGA and SGA and not elsewhere doesn’t mean it is not important. How will you know the difference?
So when should you read EGA or SGA?
A very small minority can and should and will read them as students. But you have to be thinking about certain kind of problems, and your mind must work in a certain way. Most people who read EGA and SGA as students are not in this group.
Some will read them later as they need facts, and will realize how beautiful they are.
Some will read them later for "pleasure", like reading the classics.
In summary, you should read EGA and SGA only when you need to, where "need" can have many different meanings.
In fairness, I should say how much of EGA and SGA I've read:
A small part of EGA I've read in detail. I had a great time in a "seminar of pain" with a number of other people who were also already reasonably happy with Hartshorne (and more). Reading the first two books of EGA (with some guidance from Brian Conrad on what to skip) was quite an experience --- I had assumed that it would be like Hartshorne, only more so, with huge heavy machinery constantly being dropped on my head. Instead, each statement was small and trivial, yet they inexorably added up to something incredibly powerful. Grothendieck's metaphor of opening a walnut by soaking it in water is remarkably apt.
But other than that, I've read sections here and there. I'm very very happy with what I've read (I agree with Jonathan on this), and I'm also happy knowing I can read more when I need to, without feeling any need to read any more right now (I have better ways to spend my time). When I need to know where something is, I just ask someone. And as for my students: I'd say a third of my students have a good facility with EGA and possibly parts of SGA, and the rest wouldn't have looked at them; it depends on what they think about.