Open multiple Visio documents is possible, but the default Visio behavior is to open multiple document on the sane Visio instance. In order to view each document you need to go to "Windows" toolbar and switch between the Visio documents.
You can change this behavior by simply update the Visio registry key: SingleInstanceFileOpen.
First open Visio and select Tool –> Option –> Advanced and check "Put all settings in Windows Registry".
For Office 2007 the key locate under:
Update the value from 1 to 0.
Now each Visio document will open in different Visio instance.