Is it possible to take a particle with no mass and give it mass. For example light? Or increase mass?
Give it a try. The four momentum of a massive particle is $P~=~(E,~\vec p)$ and the invariant momentum interval is $$ P^2~=~E^2~-~|\vec p|^2c^2. $$ Since E = $\gamma mc^2$ and $\vec p~=~\gamma m\vec v$ then $P^2~=~m^2c^4$ using $\gamma^2~=~1/(1 - v^2/c^2)$. Now suppose that this particle is converted into a photons with zero mass. This means $P~=~(h\nu,~h\nu/c)$ and $$ P^2~=~(h\nu)^2~-~(h\nu)^2~=~0. $$ This implies that $mc^2~=~0$, which means something is wrong.
It is not difficult to show that a massless particle interacting with another particle can generate massive particles. Also massive particles can interact in a way to generate massless particles plus massive particles.
In the case of the Higgs field, the masses of the $W^\pm,~Z$ particles is due to the absorption of the Goldstone boson that corresponds to the weakly interacting component. This is permissible. If we have two massless particles with the same energy, one with four momentum $P_1~=~(h\nu,~h\nu/c)$ and the other with $P_2~=~(h\nu,~-h\nu/c)$ then $P_1~+~P_2=~(2h\nu,~0)$. this can equal some $mc^2$. The absorption of the Higgs Goldstone bosons is similar to this.
The mass of a quantum particle is derived from the degree to which it is "linked" to the Higgs Field (a job done by the exchange of Higgs Bosons). For instance, a photon has no rest mass as it has no amount of link to the field.
In order to "assign" mass to a particle, you would have to increase how much linkage there is between it and the Higgs Field although this is seemingly impossible to do as mass is a fundamental property of quantum particles, and would be as easy to change as it would be to change the charge of a proton.
Consider it in condensed matter physics framework: we all know that around Dirac point, an electron can be considered as massless; however, if we turn on some kind of interactions, say spin orbital interaction, there would be no more linear dispersion, which give the particle somehow a mass.