I am most familiar with Match.com and Yahoo Personals, but this may apply to other websites as well. Both of these websites permit you to have a free profile, which might be relevant to the topic at hand.
I have only seen this from the woman’s side – it may be the same on the flip side, I don’t know. When searching men’s profiles, you see an awful lot where they hadn’t been active in months. At Match, they don’t give you the ability to filter by activity date, so it is especially obvious there. Yahoo has just as many, but at least you can choose to filter them out of your results. I tended to think anyone who hadn’t been active in over a month might as well not have had the profile up anyhow, but that is just my opinion.
I am aware of at least 5 men who, despite being in a relationship, would not remove their profiles from being searchable (in Yahoo terminology) or visible (in Match terminology). They would usually not visit the site at all, causing them to show as inactive for whatever the period of time was, but yet they would not hide their profile. I can understand not wanting to have to rebuild your profile from scratch should the relationship fail. However, it isn’t like making your profile invisible to searches deletes it. Both sites allow for making the profile invisible, where one click of a radio dial button would bring it back at any time. So keeping it visible is not required for anything except getting contacted by people who are interested in you.
Frankly, I think it is tacky to leave a profile searchable/visible while in a relationship – especially if you met your partner that way and they know it is there. To me, it sends the signal that the person wants to keep that toe in the water, and that if something better comes along, he/she will drop you for the better option. I actually asked one such guy why he did it, and he didn’t really answer me.
I am curious if anyone else has noticed that men leave their profiles up indefinitely (or done it yourself, if you are a guy). For that matter, do women do this too? I would tend to think women would not (unless their profile has no pictures), since the emails and attention would be very annoying. Why would you have it up just to get email that you are going to delete? Does it feed the ego to see how many people you are ignoring? Don’t you feel guilty for using people that way? And if you read them, then you ARE looking for something better, in which case maybe you shouldn’t be in the relationship you’re in. I have heard that men aren’t contacted nearly as often, so maybe they don’t get enough attention for it to be a bother. Why else would someone do such a thing? Just plain lazy? Superstitious (like a Murphy’s Law kind of thing – as soon as you take it down, you think you will lose your current relationship)? Or is it just ignorance of the fact that you can hide your profile without deleting it?
I guess the larger question is, when IS it the right time to hide a profile to your way of thinking? After a few days? After meeting in person? After a month? When you get back from the honeymoon? After your first born graduates high school?
I am not complaining about a current situation, by the way. I was reminded about this dynamic by a friend’s observation and wanted to get a more well-rounded view. As for me, I have an absolutely wonderful guy in my life right now who not only took his profile down but canceled his account the day after he met me in person (in stark contradiction to everyone else). 
He is very emotionally available, funny, thoughtful, attentive, romantic, genuine, honest, down to earth, sexy, dependable, ambitious, and all around terrific – very possibly the Mr. Right I have been looking for. Nope, I am quite happy with my situation, just curious about some aspects of the sociology going on in Internet Dating. It definitely has a culture all its own.