First off, props for a great thread. This is something that has always fascinated me to the point where I am now seriously considering writing my senior thesis on some aspect of the concept.
I'm in the same boat as the other "I'm at the stage where I know them/they know me well but wouldn't necessarily call them my friend" people, and I agree with those who say it is possible to cross the line from fan to friend (or, in some cases, remain both).
First of all, like others said, a lot of it depends on your reason for wanting to become their friend. It's good and great and all that you like their performance, but if that's all you have to go on then it's likely the relationship won't develop - like people said, friendships are about the real people. They consist of deep, emotional bonds that are on a completely different level than any part an actor might play, no matter how convincingly. It's very easy to get a false sense of an actor's personality based on the characters they play, no matter how real they seem - that's why they're actors. Not that all actors who play nice people on stage are really a$$holes in real life, but I've personally had my share of disappointments when it comes to what a performer is "really" like. Actors can be VERY different offstage, and there's no reason why they shouldn't be, as what they do onstage should not dictate their behavior offstage any more than being a doctor, lawyer, or garbageman should dictate how those people act off the job. Some of my most fascinating encounters with actors have been in situations in which there is no one around for whom they need to be "on" - in other words, no reputations to maintain, no images to uphold, etc. It really "humanizes" the actors for people who have a hard time accepting that they really are normal people (well, most of them...)
The other main point people have been making that I agree most with is that you can't just (or if you can, it's very rare) forge a friendship from nothing. You need to have some sort of foundation, some type of common ground. Thousands upon thousands of people will enjoy an actor's performance, or think they're an absolute doll at the stage door, but you don't see all these thousands of people on the phone with the actors at 4am (to use another poster's example of what constitutes a friend). These people have no shortage of others who want to become their friend. Friendships are co-dependent. Both people need an equal amount of motivation to maintain the relationship. In other words, you have to have something the other person wants - hopefully that something is a strong emotional bond or level of intimacy, the ability to confide in or rely on your friend in a way unique from all their other relationships. Of course, this takes a while to develop, but I would go so far as to say that your first couple of encounters are a good enough indicator of whether or not the friendship has the possibility of growing at all. Of course their are exceptions, as there are to any hypothetical situation. My point is that you need more than a friendly 'hello' at the stage door for a jumping off point. It's kind of a double-edged sword: as much as you need to realize the actors are just people, you need to prove to them as well that YOU are a person, as opposed to a fan. In other words, show them why your interaction with them is any different than other fans waiting for an autograph or a picture. It could be a mutual friend, alma mater, anything that would make them want to keep the conversation going, that would set you aside as an actual person whom they would be able to distinguish from the countless other people they meet in the course of their career.
I don't mean to sound like there's a certain "formula" or "steps" to building a friendship with someone of whom you're a fan - reading back over this, it kind of looks like that's what I meant. Of course every person is unique and I'm positive that there are countless examples that could disprove anything I just said - but the same can be true for how people meet their husbands/wives, best friends at college, etc. It's really something that just happens. You can only control what happens to a certain point, and then it's out of your hands. In my personal situation, for example, at this point I really can't think of any reason why this person NEEDS to keep me in their life, and if I ever called at 4am, they would probably be a tad freaked out to say the least.
Sorry for the long post (boy I say that a lot, don't I?). I'd love to hear what others have to say on this topic. Definitely one of the more interesting threads we've had in a while.
If I lay here
If I just lay here
Would you lie with me and just forget the world?