It depends on which components get mixed with which. For example, one brand's shifters sometimes aren't compatible with the other's derailleurs. Also watch out for chain widths relative to cassette gear spacing (more a concern since 2x10-speeds hit the market).

But for the most part, we're talking about mechanical (not hydraulic) devices that can be tuned/adjusted to work with each other.

It's not a problem if the parts have been intentionally selected and tuned to operate with each other. It's all springs and wire tension internally, and a good bike mechanic can make just about any (modern) groupset components work together.

However, box-store bikes are usually just built with whatever individual components are cheapest in bulk at that price bracket. More often that not, that means mixing and matching for no good reason other than to save the retailer a few cents.

Depends on the specific parts mixed. There's no question that it can be fine...e.g. see http://www.velonews.com/2013/09/bikes-and-tech/drivetrain-co...