For example, could my constraints ever give me a feasible region composed of the unit square and a triangle with vertices (2,1), (3,1), (2,3)? (Edit: obviously this would never happen, but I am using this to illustrate what I mean by 'parts')
My immediate answer is no, as this would mean I have conflicting constraints, but I'm having trouble articulating this...
The overall feasible region is the (set theoretic) intersection of the feasible regions corresponding to each individual constraint. If the constraints are convex, then each individual feasible region is a convex set, and it's fairly easy to show that an intersection of convex sets is again convex. A convex set can not have several disconnected pieces.
In linear programming, specifically, each individual constraint is convex, so what I wrote above applies to your situation.
What you said about conflicting constraints isn't quite correct. If your constraints are conflicting, then your feasible region will be empty, so it doesn't make much sense to ask whether or not it's convex or connected.
