For planar graphs, this theorem follows from flow/coloring duality, and the Five color theorem (every loopless planar graph is 5-colorable). In light of this, we may view this conjecture as a widesweeping generalization of the 5-color-theorem. The Petersen graph does not have a nowhere-zero 4-flow, which shows that this conjecture (if true) is best possible.
It is far from obvious that there should exist a fixed number so that every bridgeless graph has a nowhere-zero -flow. Indeed, this weaker conjecture was also made by Tutte, but was resolved by Kilpatrick [K] and independently Jaeger [J], who both proved that bridgeless graphs have nowhere-zero 8-flows. Seymour [S] improved upon this result by showing that bridgeless graphs have nowhere-zero 6-flows.
[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216. MathSciNet
[K] P.A. Kilpatrick, Tutte's First Colour-Cycle Conjecture, Thesis, Cape Town (1975).
[S] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet
[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet
[Tt66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet
* indicates original appearance(s) of problem.