Techincally you can combine any martial arts, but it will be difficult to excel at either of them, unless they're really similar.
Different arts often require opposite reflexes which can (will) be confusing, especially for beginners. My advice is to start with one and after you reach a decent level, choose a second. That way you can place the differences and know when (and mostly WHY) you shouldn't use the basic stance from MA 1 in MA 2 and vice versa.
I actually followed an interesting workshop for MA trainers this weekend about teaching kids basic positions (both for standing up and ground work). There were karate, judo, JJ, BJJ, boxing, ... trainers. The teacher was a BJJ guy.
Some reactions from myself and other trainers there clearly show my point:
Exercise: When avoiding a blow (or neck grab), the practicioner had to duck under the arm, go behind his opponent, grab him around the chest and keep him very very close, leaving no space between any part of his body and the opponent's back.
This is how we looked at it:
Teacher (BJJ): if your opponent is behind you, you're in trouble. NEVER show your back to someone. The guy ducking and grabbing from behind is now in the best position.
Me (judo): having an opponent that steps behind my back me makes my fight A LOT easier, I don't have to pull him there or turn, I can just lift my leg, twist my shoulder and throw him. Thanks for the free score.
Boxing trainer: ducks and steps nicely, grabs opponent with arm, but leaves a huge space between him and the opponent's back, standing far away with his legs and hips. -> boxer's are not supposed to get close to each other, it's against his reflexes.
Karate trainer: I'd never get that close to someone, let alone grab them closely.
In judo (and BJJ) getting close to your opponent is absolutely necessary. And it's difficult to learn. It usually takes years for a young judoka before he automatically brings his hip close to his opponent. This behaviour however, is not good at all in karate, taekwondo, etc.
Also in judo, you're feet should be more or less on one line (usually one stands a bit in front), parallel to your opponent. In karate and taekwondo, for example, you don't align parallel to your partner. You show so much of your body then, so many space to hit... those fighters will stand with one side far away from their opponent and one foot very clearly in front (= free foot sweep for judoka).
In BJJ and JJ laying on your belly is dangerous. You're weak and have no defense. Losing is almost inevitable.
In judo, on the other hand, laying on your belly is a very safe and sturdy defense posture. You're not in a position to attack, but you can hold off attacks very well.
Learning to defend well laying face down takes practice. Most judoka that have been practicing for only a couple of years, I can easily turn to their back or get an arm free to lock. Same for the stance with the feet on one line. Most are "afraid" in the beginning and will try to keep one part of their body away from you. Leaving a free foot to sweep. This takes practice.
Now imagine combining the stand up work from karate and judo and the groundwork of judo and BJJ. It gets quite confusing and it will be difficult for the positions to become automatic. Every time you have to think "now I should do it", "now I cannot".
And if you need to think about your position during a fight, well, you've already lost...
So basics from different arts can really contradict each other. Although it all sounds simple, it takes a lot of practice for those basic stances/positions to be automatic, believe me. Trying to mix them will only lengthen this practice time and you might never get the real hang of either of them.
Depending on the goals you have with your chosen MA's, this will or will not be a problem.
If you want to know a lot of different techniques and defense/attack situations for real life, combining e.g. judo and karate is great, because they complement each other as stated in the answer above.
If you want to be a very good judoka or karateka, you won't fare well by combining them. You'll lack the automatic moves in both to become a champion.