Is there a limit to the number of lightbulbs that can be controlled by one hub? If so, how do I increase that?
Tens of thousands, theoretically, perhaps hundreds or maybe thousands in any practical application, so ... yes, but probably not anything you're running into. 
(This is the number of Zigbee nodes you can have on a single network, not anything related to bulbs per se.)
One thing you might run into is the number of end devices that can connect directly to the hub. This is 32 on most hub models, maybe all? I can't remember if it was increased to 64 on the newer ones. But this isn't a hard device limit. End devices are non-routing (non-repeating) nodes, but most bulbs are going to be repeaters. Sengled is a notable exception and the only one I'm aware of. And it's easily worked around by adding routers to your network, each of which will have its own limit but extend your capacity by whatever that is -- and more for each router you add.
If you're having problems, I'd share more information:
- What hub model are you using?
- What Zigbee channel and (if C-8 or newer) power?
- Any other Zigbee networks, 2.4 GHz Wi-Fi networks, or anything that might interfere in the area?
- Brand and model of bulbs? Have you made sure they are reset?
- What other devices (brand and model) are on your Zigbee network so far? The Settings > Zigbee Details may be helpful to figure this out.
- What exactly happens when you try to pair a device?
Only the C-8 Pro supports up to 64 direct devices, all other 32.
The DLV procedure has been known to be helpful with troublesome ZigBee devices. 