Is there a good rule of thumb for how many lights you need in a room?
You need to consider many factors.


Beam angle: spot (narrow) vs flood (wide), and everything in between


The wider the angle, the farther apart the spacing can be without creating dark spots on the wall

Direct vs indirect light: is the light facing the wall, so the room is actually lit by light reflecting off the wall, or are the lights directly lighting the floor/whatever it is in the room? 


Reflecting off the wall will make less lights seem brighter (vs direct lighting), and creates a softer light (not many shadows, light coming from all directions)

Wall color: especially if you're mostly using indirect lighting from the wall, the color will dictate how much light is reflected


Light colors (eg, white) will reflect the most, while dark colors (black/gray) will reflect less. 
Strong/vibrant hues (eg, red, blue) will also cause the room to appear that color. Eg, if you have red walls, and all your lighting is indirect from reflecting off the walls, the room will look like it's lit with red lights. If you use direct lighting in the same room, objects/people in the room will have closer to normal colors (you'll still have some element of red, of course)

Heat: depending on the types of bulbs, more heat will be generated, which should be a factor to consider. A room with a dozen halogen bulbs, for example, will be noticeably hotter while the lights are on full. CFLs and LEDs generate much less heat, but have other drawbacks.
Dimming: If the lights are dimmable, and you plan on having them dimmed a lot of the time, you may need more fixtures to generate an even look. 
Electrical load: Code says you should design to use 80% of a circuit's capacity. If you have a dedicated 15A, 120V circuit (breaker) for these lights, you can use up to 1440 watts total. This means if you have 100W bulbs, you can only have up to 14 of them. 


If the circuit is shared with other lighting/outlets, be aware that subtracts from the power you have available. 
In Canada (at least) there is a limit of 12 outlets per circuit. I'm not sure if that applies to light fixtures or not though, maybe someone else can weigh in?