The entire package probably cannot handle that current for long. You should choose the resistor values so that R+G+B current combined does not exceed the maximum rating of the package itself. Even if a single part separately can handle 20mA that does not mean the whole package can handle 20mA x 3.
There are more intelligent/efficient ways to drive LEDs than just resistors, such as constant current drivers, which may be controllable in a way that it could dynamically adjust current per LED segment based on other inputs. Like when only R is on, set R to do 20mA, but when R and G is on, set each to only 15mA to respect the package ratings etc.
Edit: If you look in the datasheet for a "maximum ratings" table, it will usually specify the total "power" that the package can handle. The power can be estimated from forward voltage drop of each LED for a particular forward current, which you can find for each of the different coloured LEDs.