Apple - Why can't I drag columns in Finder's Column View?

This happens when that separator needs to accomodate space for a scroll bar due to content length. You can resize them with the handles at the bottom end since 10.0.

I can confirm that Lauri Ranta's solution above fixed this issue for me.

System Preferences -> General Preferences -> Show scroll bars: "When scrolling"

Like Andrew mentioned, this problem seems to come and go. On my machine, I wonder if the setting "automatically based on mouse or trackpad" is getting confused as I switch between a Razer mouse and the Apple trackpad. Might be worth a look if you have a similar setup to mine.

Have you tried changing the setting for showing scroll bars in the general preference pane?