No...we are talking about the rest mass here...not about the total relativistic mass...photon, for example, is massless...
in the electroweak theory, electrons and photons are originally massless...It is something like this..the particles have a lagrangian...there is a definite form for the mass term of a particle in a lagrangian...but in the electroweak lagrangian those terms are not allowed as they break the symmetries we want the lagrangian to have in order to explain experiments...so people found another way to bring about the mass terms...but that method requires at least an extra particle called the Higgs particle..
Neutrino and the photon are massless even after all this...but again, mass means rest mass only..