I remember getting some flack here a while ago, when a friend and I were planning a trip together- I love high up places, and my friend is somewhat acrophobic.
Somebody said that they would be VERY offended at the idea of splitting up when traveling with somebody, and would have bad things to say to and about the person who proposed splitting up.
But I'm with most of the posters here- I think it's better to split up sometimes. If one person is tired, sick, or otherwise not well, there should be no hard feelings if they want to go back to the hotel early and rest for a while, while the rest of the group continues on. If I were part of that group, I would want to be sure that the person bowing out was comfortable enough navigating to get back to the hotel safely, or could grab a taxi. I wouldn't want my sick or tired friend to get lost after all!
Splitting up also allows people with different interests to both be satisfied without boring the other person. Say one person is a comic or gaming geek, the other loves cookware. In that case, if they insist on sticking together, one person is going to be enthusiastically browsing a store specializing in their interest, while the other person is bored silly. So perhaps splitting up is a good way to preserve the sanity of both partners.
Having to do everything together like a line of ducks when traveling would annoy me too!