There are two issues which you need to consider here:
1) What are you trying to do? If you are trying to be a high-level competitor in two completely separate standup arts, this could be very difficult. you'll have to learn nuances of both, and be able to switch between them and keep them separate. I'm not sure how well you can do this without getting confused or having things merge together.
If, on the other hand, you are trying to round up your game and take from two separate standup arts to form your own style, this is more likely to work. Similarly, if you have one main art and want to take another one to supplement it, then it can work. You have one main mode of fighting, and add techniques from the other style and incorporate them.
If this is what you are trying to do, I recommend going in with a very open mind and emptying your cup. Completely forget the other art and do everything exactly the way you are taught. This will help you understand the art in the whole. Don't worry, your previous experience and knowledge in the other art WILL eventually transfer over and you'll end up with your style combining parts of both.
You will need to be careful, and do some thinking about what your fighting style should be like, what you want to do. I recommend letting your instructors know that you are cross-training, this can be a great help. But definitely be respectful and do as you are told. Nobody likes smartarses who think they know everything because they trained something else.
2) The second factor is how well the arts fit together. Even more precisely, how the styles you are being taught (there are many different ways to box and many different ways to kick) fit together, and how they fit into what you want to achieve.
As people have already written, some arts with a lot of overlap can be hard to fully integrate. Some karate styles are very stiff and straight-forward with their attacks, giving everything into every strike. Other styles, like TKD, some boxers, are very fluid, constantly changing angles and using many combinations. It can be really difficult to learn both at the same time, and it is impossible to fight in both ways at the same time. You'll have to make some hard decisions.
As for my experience, I have always been a proponent of getting quite good at one art, making it your base, and then branching out. This way you can be very successful in adding missing tools to your arsenal.
Other people find that learning several arts at once will develop your own "style" sooner, like MMA fighters do. But I feel that in this case, you have to have the RIGHT combination of arts (and I personally don't think Karate and Muay Thai are the right combination for this) and have trainers who know how to combine them effectively. MMA is an example of this with people learning combinations like Muay Thai + wrestling + BJJ.
Karate + Muay Thai is a rather difficult combination, IMO. They use pretty much the same movements, only differently. I don't see too much gain in combining them and lots of headaches. If you are proficient in Karate, I'd suggest some crosstraining and adding some knees and elbows and clinchwork to your skills. Otherwise, I'd pick one and stick with it. Depending on your goals, Kyokushin karate OR Muay Thai could both be a good base.
Then, if you want to cross-train, take something with little overlap, like Judo or wrestling. You'll probably have more success and fewer headaches.
how can u guys afford so much training??
