Yes, you can boulder by yourself (I find bouldering more fun anyways). Most gyms have sign up sheets to connect solo climbers with others. Even better, when you sign up at a gym you'll probably receive guest passes, take a friend some time and they'll probably want to sign up themselves because its so fun. My group has increased from 2 to 4 over the last few months.

EDIT: For those who are unaware of the climbing terms, here are some brief descriptions:

Bouldering: Climbing solo, with no gear, usually up a wall ~8-12 ft (with mats below, in case you fall). These routes are usually tougher focus more on technique/strength rather than endurance.

Top-Roping: Climbing with a partner (one climber, one belayer), both partners must have the proper gear and tie in to the ropes in order to climb. These routes are usually 20-50ft and require a good deal of endurance to reach the top.

I found it very straightforward finding a partner to climb with when I started. Rock climbing gyms are very friendly places in my experience, and as others have said, most will have some kind of system for pairing up people who are looking for partners.

I'd suggest dragging a friend along to your local climbing place so you can try it out a couple of times and see if you like it. Then if your friend doesn't want to keep going, either speak to the staff at the gym, or if they can't pair you up with someone put an advert on craigslist/gumtree/similar (I got 3 responses to a gumtree ad when I started, which gave me an instant group of people to go with).

As long as you can commit to a regular time once or twice a week, you shouldn't have a problem finding someone to climb with. Once you start going regularly you'll meet other people at the gym, which gives you more pairing options if your normal partner is away or ill or whatever.

Try it. I'm glad I did. I've made some new friends from it and gained a lot of strength.

Just be careful bouldering as a beginner. It's really easy to try something that you don't have the hand strength/tendon strength for yet and tweak something. Some gyms have autobelays which let you climb without a partner, so that might be an option.

I would STRONGLY recommend AGAINST bouldering to start, because of the HIGH likelihood of injury from falling.

If you are not used to taking impacts from jumping from high places (e.g., typical sedentary person), you will almost certainly sustain a severe injury.

Broken and dislocated ankles and ligament ruptures are alarmingly common. Dislocations are especially bad, because the ankle's blood supply isn't great to begin with, so a severe dislocation is like yanking a plug out of the wall; if the blood supply doesn't happen to get re-established (which even microsurgery can't do anything about), you are looking at avascular necrosis. Imagine a dead coral reef crumbling away.

It's even worse in climbing shoes because your toes are crammed together -- imagine landing on your fist.

Rock-climbing (roped) gives your feet and ankles time to strengthen as you support your weight on tiny toe-holds, which is a slow process because ligaments and tendons have poor blood supplies compared to muscles.

Looking at elite climbers bouldering is like looking at elite gymnasts -- you don't see all the ones who got seriously injured along the way. You see them taking 20-foot highball drops and think, how bad can 10 feet be? Or 5 feet?

Even a drop of a few feet can result in a severe injury if your foot isn't flat; when you are jumping off backward onto a lumpy bouldering pad, if your foot lands inverted slightly, the impact can force it inward 90 degrees.

This probably happens twice a week at popular climbing gyms with bouldering areas.

I'm going to slightly counter brildum's statements and point out that I was unable to stay on a bouldering climb for something like six months after I started top-rope climbing. I just didn't have the strength (less important) and skill (more important).

Admittedly, I have known a few people who've gone straight into bouldering and pulled it off. Give it a try. If you can, great.

By the way, one of the best things about rock climbing is the social aspect. You should look at partnering as an opportunity rather than a hindrance. I've made a lot of good friends at the gym over the years.