I also want to add that she was tested for a urinary tract infection; however, the vet was not able to get enough urine from her to conclude with certainty that she had a UTI. As a precautionary measure, they provided me with the necessary medicine to treat her for a UTI as a precautionary measure and that treatment process was completed by the end of November 2010.

There are many reasons a cat pees on furniture, etc., so I've included a link below that may be of help. Notice that it mentions you should have a litter box for each cat plus a third one.

I would first take her back to the vet and have another test done to rule out UTI or crystals. Personally, I would have them run a blood panel to check her kidneys and any other possible problems.
If your vet isn't able to draw a urine sample using a syringe/needle then find another vet. You may already use a clinic that specialize in cats, but if you don't, I highly recommend you try them if there's one in your area.

Questions: Are you sure it's always Muggs that is the culprit?
How old is Muggs?
Is it always in the evening when it happens and are you always in the room?
Can she access a litter box without having to pass by the other cat, and have you tried placing a litter box in your bedroom?
Are you gone from the home for long periods of time? I ask because I see your user name has 'pilot' in it.

You may also want to check out this article: http://www.stopcatpeeing.com/
I have no idea if it's worth reading!

Here is a link that might be useful: outside the box

I had a similar situation recently,although it was not on my bed that the new cat peed. We found a stray cat who we SLOWLY introduced to our existing pet family ( 1 adult neutered 3yr old male cat, and 2 adult neutered dogs). The new found stray ( @10 yr old neutered male cat) started urinating on the carpets in my 2 daughters rooms. After $hundreds of dollars worth of testing that showed NOTHING medical,and litter boxes being cleaned hourly- it was said to be stress. We put him on Fluoxetine ( prozac) and it stopped COMPLETELY!!! This medication saved his life and mine :o). It is very cheap- $10 per month.
If nothing medical is found, just keep this possible solution in mind, and good luck with your new kitty.
Also, here is something I used to get the urine out of the carpet-
16oz peroxide
1 teaspoon dish soap ( ie dawn,palmolive)
1 tablespoon baking soda
blott up as much urine as possible, mix ingredients togetherand DUMP all of the mix onto the spot. Let it air dry. This REALLY worked for me.

I don't know what your situation is, but if you could just keep kitty out of your room and keep the door shut until he develops proper peeing habits, that could be all it takes. He's the new guy and a little confused about things.

And he probably does need to be able to use a box where Chloe can't glare at him.

Been there, done that. With my cat, it was 100% a territorial/behavioral issue. He didn't get along with our other cat (they fought constantly), and whenever he was feeling stressed or especially territorial, he would pee on our bed. Here's a few tips:

1) Get an enzymatic cleaner, like Nature's Miracle. You can pick it up at any pet store or online. You need this to break down the odor smell, both for your own sanity, and to help prevent further accidents in the same spot.

2) Get a waterproof mattress cover. You should easily be able to find one for around $30 -- take a look online. Trust me, it's a lot easier to deal with urine on the bed if only the sheets/blankets are affected.

3) Have at least 3 kitty boxes, and scatter them around the house.

4) As homebodymom mentions above, look into putting your cat on some sort of medication. My cat was on something similar to Prozac, called clomicalm. It worked miracles for us. Only downside is that it made our kitty sleepy and less active, but we were willing to live with that.

We were able to take our kitty off the Clomicalm once our other cat died. We haven't had a single accident since then -- that tells me that it was definitely some sort of territory/marking issue caused by the other cat.

Just my opinion here, people. Cats are territorial and dislike any change in their environment. I always let my cats go in and out over the last 40 years that I have owned cats (or they owned me.) In most of those years, I never had a in-door litter box. They always let us know when they needed to go out and usually spent nights outdoors. Indoor cats very simply have 'issues'. They weren't meant to stay indoors, especially at night. Cats are nocturnal.

Over those years there was lots of change in our lives --- kids going off to college, etc. Our cats adjusted and sometimes just disappeared when they were stressed. No inappropriate peeing. No problems. When life in our house adjusted, the cats came back and their lives adjusted.

It is all so simple if you let the cats be cats.

I have had indoor cats for 40 years and I have never had a cat that peed inappropriately. When I have had multiple cats, I went by the "box for each one plus one" theory. I agree that stress probably does play a part in this, once a physical problem is ruled out. Keeping the litter box clean, and giving the cat lots of attention should help. If you can figure out what's stressing him and remove that, that should help. Just make sure it's not a physical problem. A urinary tract infection can be really painful for a cat. Good luck with this. Let us know if you can figure out what the problem is.

I suspect the problem you are having is strictly a matter of territorial competition between the cats. Your alpha cat is dominating the newcomer and not allowing it to use the litter box. So it is forced to void in the wrong places. I think it will be a persistent problem. Although you may reduce its frequency or severity to some extent, it will persist. Some cats are unwilling to accept a newcomer into the house and will make the newcomer fearful, unhappy, and will drive it to this kind of problematic behavior.

Your alpha cat wants to be a loner.

I have 2 indoor cats, they r both 3 and although they are from different litters, they are like brother & sister (they r both de-sexed) so u don't have an alpha etc.
Recently we moved inter-state & sprayed the new house with feremones to make sure the cats weren't stressed etc... That worked well, we have now been here 6 months & my girl cat has taken to peeing on our bed! She did have a UTI, which was promptly treated, yet she recently has done it again!
I need help as I don't know why she is doing it? She uses the kitty litter normally and in the 3 yrs we have had both cats, we've never had an accident??
I cant afford to keep replacing my bedding (as I cannot get the smell out) or sleeping on those plastic mattress covers! She is so naughty now as well, doesnt listen to me anymore & scratches everything!

Any suggestions would be most welcome!
Thanks