If your RGB LEDs are WS2812-style, 5050-size units, it is no problem to control dozens of them using an Arduino Uno or several other models of Arduinos. [For ATtiny systems, other libraries than Adafruit's NeoPixel may be needed; the NeoPixel library keeps several bytes of state data per pixel, which some other libs don't.]
WS2812-style LEDs typically are arranged in serial strings, driven by a single IO pin from an Arduino.
If you have more than half a dozen WS2812s in the string, power for them should come from an external power supply, rather than via an Arduino's built-in 5V regulator.
For more information, see related questions like Arduino with WS2812 using Adafruit NeoPixel library fade in/out different patterns, and Arduino and WS2812 RGB LED Strip "freezing", and fading colors on a ws2812 strip; also see Google search Arduino Stackexchange for WS2812.