I love bike touring, and have done a couple of multi-day tours with others, but I've only done overnighters by myself. Both ways have their advantages and disadvantages, but the feeling of freedom and self-sufficiency when you're on your own can't be beat. As Cassandra says, you have to know what you're doing, mechanically, or at least have done your homework so you know where to get help with bike problems if you need it. Also, lots of research on your planned route--I actually find this part exciting and fun, kind of like the appetizer before a special dinner.

And yes, DeniseGoldberg comes to my mind, too, as a person on these forums who's done her share of solitary touring. In fact, she's bike touring in Nova Scotia right now, and you can read her tour journals at CrazyGuyOnABike.
Also, look for her posts in the Adventure forum.

To me, if there's a choice between going alone or staying home, I'm definitely going alone.