Connection examples and embedding dimension theory

Question

I've read about the "Utility Problem" (i.e. three utilities and three customers) requiring three dimensions to accomplish/attaching/embedding; and the Klein bottle requiring four (space-like) dimensions. Presumably this relates to embedding.
My question is: how to prove that particular connection/attachment problems throw the "manifold" into higher dimensions. The example of the Klein bottle has a two-dimensional domain/surface, but pushes the embedding space into four dimensions.
References are fine, since I am trying to learn and have a particular problem in mind; and it would seem to require detailed knowledge. My background is one quarter of point-set topology, which doesn't seem to cover this.