Actually it would be entirely accurate. Mass of a particle is like an avalanche, and the small "seed" mass given to the particles by the Higgs field is the first snowball that gets the avalanche going, or the match that starts the fire. All other interactions builds up on it to make the particle's mass many times that of the original Higgs given mass, but that one is necessary to begin this chain.
Thank you sayak83,
This passage points to mass contributions from other than Higgs mechanism. In case of electron, what percent of its mass is contributed from those other than Higgs mechanism?