I don't know about cats (I haven't tried training) but my dog definitely knew a few nouns and verbs. She understood "food", "water", "walk", "bone", "ball", "bear" (her toys), and could distinguish between "point", "fetch", and "drop". With "fetch ball" she would go get the ball, whereas with "point food" she would point (paw) at the food, and with arbitrary combinations of these verbs and nouns.
It's astonishing, I didn't think they could do that, but apparently they can.
In our case:
over there; bed; basket; lead; walk; wait at the door; go to [room]; go to [family member]; stay; stay on the path; paw; paws up; sit; lie down; heel; don't pull on the lead; no more poo poo; no more wee wee; this is the commute; up; where's your piggy?; treat?; buscuit?: up for a cuddle?; want to come up?; permission; okay; no; stop; this way; stay close; hungry for some food?; water?; down; don't lick; who did this?; come here; shall we go the pub; let's go;
and probably 5 or 6 more words/phrases. I don't include words like catch/fetch where accompanying action may trigger the response.
It's definitely not a three year old's grasp of language, but it's probably a point that a baby crosses somewhere between 9 and 15 months old.
I have two cats that I have clicker trained. They can: shake, high-five, fist bump, stand up, and will "go here" and come and touch your finger. They won't do any of the tricks without treats :)
One of the cats had seemed interested in fetch for several years but it never quite worked. I tried a bunch of different toys and finally found a plastic spring that she will fetch very well.