Adding luminous intensities (cd)
Lumens DO add directly.

If the illuminated area is common to two sources then
Lux (luminous intensity) = lumen per square metre also adds linearly.

If you has say 3 LEDs (each of say 1200 lumen output), each exactly and uniformly illuminating a 1m x 0.5 m area and you placed the areas side by side to make a single 1m x 1.5m area, then  


Each area would have 1200 lumen of light shining on it  
The total light would be 3600 lumen  
Each sub area would have a luminous intensity of 1200/0.5 = 2400 lux   
The 1.5 m^2 area would also be illuminated at 2400 lux (  (3 x 1200) / (3 x 0.5) )


In practice LEDs  do not illuminate an area evenly so the lux level will vary across the 0.5 m^2 areas and across the overall 1.5 m^2 area. If light from a given LED falls outside it's 0.5 m^2 area (as it probably would) then at the boundaries with the other areas it would add to the light from the adjacent LED so the boundary areas would be brighter when the 3 areas were merged.

In practical situations where you have more or far more than one LED per 0.5 m^2 the level of illumination would be more even than with only 3 LEDs.