May I be a bi-ritual Catholic, meaning both a Roman Catholic and Byzantine Catholic?
Each member of the faithful is a member of a local church, under the authority of a prelate of that local church. So, in fact, it is not possible to be a member of two different local churches at once.
However, if your own circumstances require it, for reasons of marriage and family or of ethnic diversity among Catholics (as for example in the Middle East), and you regularly assist at liturgies in a rite other than your own, this is perfectly fine, and there is no problem with it. Even so, you remain a member of a local church under its bishop, even if it is a rite you do not always follow.
This is not so much a problem when the person is of the Roman rite. But it can be more of a problem when the person is of the Eastern rite, since they often run the danger of losing their people to the much larger Roman rite, especially in countries to which they have immigrated. One should strive as much as possible to maintain the Eastern rite of which one is a member.
For Roman rite faithful, Eastern rite participation is an enrichment, but often enough for Eastern rite faithful, Roman rite participation is an impoverishment. Small is beautiful, as the saying goes, so it is good not be lost in the larger rite but maintain fidelity to the rite of one's inheritance.