Hoping to speed traffic through an increasingly congested Internet, several big Web companies including Google Inc., Microsoft Corp. and Facebook Inc. are paying major broadband providers for connections to get faster and smoother access to their networks, say people familiar with the matter.

Netflix Inc., though, has held out—so far. For a year it has been trying to directly connect specialized technology to the networks of broadband providers as a way to improve the quality of its video streaming, avoiding the stops and...