I have the first-gen XPS 13 with 1366x768 internal display. No problems at all in a multi monitor setup. I have a Dell S2409W, 1920x1080 at home and a NEC 1920x1200 at work.
> Do you use both the internal display and external monitor?
Yes
> Have you tried connecting more than one external monitor?
Never tried. This laptop has only one mini DisplayPort output. I heard that there are splitter cables/adapters which should allow for that but I haven't found an affordable one yet. An alternative would be a display that supports daisy chaining (the only one which I am aware of is Apple's Thunderbolt display which doesn't even turn on for me).
Have you tried connecting more than one external monitor?