Yes Lauren, each plant will only flower once so the mother plant is only good for more pups once it has flowered. It is best to remove the pups as soon as they have their own roots, or are 2/3 the size of the parent plant. The reason to do this is that the parent will try desperately to produce more pups if the first ones are removed and each parent has the potential to produce a pup from the eye which is dormant beneath each and every leaf on the plant. In reality you wont get a pup from every leaf node but as long as you can keep the parent or its stem alive there is a chance of more pups. By carefully manageing this potential you could get between 5-10 pups off many of your broms which is far more than they will produce in a year if the pups are left on.

Only difficulty is finding somewhere to put them all! :)

Yes I have that problem too, its ok when you can give them away.
But when they have to go in the compost or the wheely bin its a shame.

As a general rule, pups will take 2 years to flower. I often don't remove them from vriesias etc because they clump out just fine & when the parent is dead the old plant can just be twisted out. Also (erhem) sometimes because I'm just plain lazy. Also Guzmanias take a little knack as sometimes you can kill the whole lot!
Neo's are better seperated because the sheer size of the mature plants may cause the pups to grow lop-sided or deformed (except the stoloniferous ones).
So, being guided by the rule stated by the poster above - to do so when they have a little independent root system (*) or are 1/3 size of parent. Guided by the size of the flowering plants, allow sufficient space when re-locating them, for them to grow without overcrowding. Think what a glorious show you will have in 2 years or so. Place your taller varieties at the back and the littlies in the front. Alternatively plant them in pots and you can move them around. I use 1/2 & 1/2 ordinary potting mix and cymbidium orchid mix with a bit of slow release chucked in. Many of the smaller varieties will do fine in squat pots but make sure they are wide enough to accommodate the root system up to maturity, unless you want to be forever re-potting. Taller and heavier varieties, can go into deeper and larger pots, with a layer of stones in the bottom. That will help a little with drainage but also add some extra stability to stop them falling over. Another little trick with pups - even if they have some roots - when planting out or potting up, pop a few stones around the base of the plant on the surface of the soil - this will stop them toppling over while they get established and can be removed later.
(*) Incidentally I have bought many offsets from professional growers which had no root system but were cleanly removed, they did subsequently form roots so long as placed in well-draining mix and I cannot remember having lost any that way.
It is true that removal will often force parent plants to throw more pups but sometimes these are not as strong as the first offsets. It's up to you though, especially if the parent was especially nice and has "had it's day" it's worth plonking it somewhere out to pasture to see if anything more eventuates.
Oh and don't do what I do, that is, *THINK* I'll remember what they were. Little broms look awfully alike, label them, even if you don't know what they are, with SOMETHING like "red spike" or "orange cup" or "the one Mum and I bought at the markets" LOL

The first flower has recently appeared on one of my broms, and its an absolute gob-smacker. I didnt plant them for their flowers, as I know they die after flowering/reproducing, but this amazing rosette is just gorgeous. More a thick, scarlet cone than the brom flowers I'm used to seeing, and all the more spectacular for the fact that the broms were largely discards - left over after some tree clearing at the front of the house. I'll get my act into gear and post a photo - it may well be the first thing in my new garden worthy of publishing to a wider audience :)

Cheers,

Artie