It'll happen, but the vast majority of web sites won't be able to be successful off of this business model because...well...there's a lot of crap out there that people are willing to glance at for free, but will not pay for if they must subscribe to it.
edit
For example, if this were to happen to computing web sites, I can honestly say that I only see a handful of our beloved hardware sites remaining - AnandTech, Tom's, Sharky's (unfortunately

), Ars Technica, and possibly HardOCP.
edit #2
This biggest stumbling block for subscription services is the fact that one person could subscribe to the service, and split the cost with X number of other people, providing them with the login/password to the web site. In fact, I can see this very thing happening with Napster. And how do you prevent one person setting up a proxy server that has access to the subscription server, and then allowing people free access to the subscription service through the proxy. There are many ways around it, and I suppose you have to rely on people's honesty. But don't expect these subscriptions to be anywhere near the level of paper magazines' subscriptions. I can see myself spending no more than $2.50 a month on my most frequented web sites. I already pay $40 a month for internet access, so any subsequent costs start to add up.
-GL