Now that's a great question indeed! Evidently, at 0K all elements except helium are solids, at 10000K they are all gases, so someplace in between the number of liquids must reach a maximum; what and where might that be?
Well, there is no formula or theorem that says liquid hydrogen must boil at 20K, nor is there such a thing for any other element, so this line of thinking hardly provides any insight. Let's try another way.
Let's take a look at all elements' melting and boiling points. (I used the great site referenced in a comment by ringo, where all data are conveniently stored on line 47 in the file http://www.ptable.com/Static/interactivity-2fcd37b.js). Let's simply make a joint list of all melting and boiling points, sort it (keeping track of which is which), and then walk through it from bottom to top, calculating the number of liquids on each interval. If you know some kind of programming, you can do it with a script.
It turns out that the maximum number of simultaneously liquid elements is 49, and it is reached several times between 2183K (melting of vanadium) and 2792K (boiling of aluminium). By dragging the slider you may actually see which elements are included in these 49. Now, the data are not terribly reliable (for example, in higher actinides they claim some melting points obtained by God-knows-what extrapolation, but no boiling points at all), yet still this is something to think about.