are there limits for the number of buck converters I can run off a single power supply
I have had boards with 12 different power rails, including two negative (half of the voltages were dedicated to an LCD), but not as much current.

You have 240 watts to begin with.  Total up the values for the buck converters, perhaps conservatively using only 70% efficiency, and you get 

$${(19.2\times 0.7)\over 0.7} + {2\times (19.2\times 1.4)\over 0.7} + 24\times 1.05$$

and you get a total of: 

$$19.2 + 76.8 + 25.2 = 121.2 \space watts,$$

which is about half of your budget.  So you should be okay.  I wouldn't go more than 3/4 of your maximum, or 180 watts.

Be sure to follow the datasheet's recommendations for whatever caps you may need at the input to the buck converters, and locate them close to the IC's.