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In standards compliant browsers (read: forget about IE6 for now) setting a height means fixing it.
If you then add more than can fit in it overflow will dictate how it all will react to "I've 5 moneys and only room for 3". Either the 2 extra monkey can swap cages with the other 3 as needed (scroll), either we release 2 of them in the wild and they occupy the neighbor's garden (visible), either we hide them out of view (hidden).
If you don't want the box to be fixed in size, well then don't set the height.
If you want the box to have a minimal size, set the min-height (to e.g. show 3 cages even if we currently only have one monkey, so we show 2 empty cages).
Similarly max-height can be use to set a maximum.