Sure there's such a thing as a depressed cat! I have one of my own, he's also got anxiety issues.
Not every cat will like the company of another animal, it's just how it is, others will opening accept another animal with time, and some will always have little tiffs here and there, but in general a "peacefull" co-existants.
If you want more then one cat, it's normally better to team them up with a new buddy(s) when they are very young. Older cats are set in their ways, and frankly an intrusion such as this can sometimes make them grumpy if you will.
I have noticed, with my Asim, any additional animal (specially cats or dogs) that have come into this house (I foster) his personality changes a little bit, and in some cases it changes when that said animal leaves as well.
He tells me when he's sick of having so many animals in the house, or when I'm not paying him enough attention, etc... and during those periods I do stop fostering.
Make sure you pay plunty of attention to your first cat, give him a place he can retreat to if he is feeling over whelmed, such as a section of the house just for him, or a nice big room where he has toys, litter box etc.
Also, did you do proper and slow introductions, this part can be very important! How long have you had this new cat, it can take weeks or months for some cats to finally get along. Give it time and play with both of them together a lot.
If things start to get much worse take the old cat to your vet, perhaps some behavoiral medication is in order (I personally prefer not to do this with Asim) but maybe they will work for you cat.