This happened to me when I was interviewing for graduate schools—two different schools offered only one weekend per year, and picked the same weekend. However, a number of students had the same problem, and contacted them. As a result, the departments in question agreed to "share" the weekend, with part of the time spent at one school, and then part at the other.
If the schools are not geographically close, and you have no way to split the weekend in such a manner, then you should contact the schools in question, to let them know that you have a scheduling conflict. In some cases—particularly if the department is large—they may be able to schedule you to visit on another date. To some extent, such a visit might be even more useful than the scheduled group visit, because you get to see what the department is actually like when they're not trying to impress everyone!