If you hit the n key you'll get a little box that lets you move things numerically, which can help you line things up exactly.

You can also select one vertex, hit shift-s and either snap things to the grid or move the cursor to the selected vertex and then move the other vertex to the cursor.

You can also increase the range of the "remove doubles" command.  In the buttons panel with the edit buttons selected (F9), one of the button panes has "Mesh Tools," and there's a button for remove doubles.  If you raise the number in the limit field it will remove an extra vertex when two of them are further away.