multiple dropbox instances hangs in starting loop
Personal Dropbox does not work in this fashion, as it causes open preference file conflicts, thus the hang on the startup of the second instance. You must login and logout of each personal Dropbox account to use them individually, or used shared folders for common files. You can also have two Dropbox accounts open in different web browser windows.

If you are a Dropbox for Business user, you can have personal and Business accounts sync to single devices, which is what you may be looking for.