The system won't allow it, but that doesn't mean it cannot be changed. The question that remains is what is needed to change what we have.
The bicameral system we have in place was not as originally intended. It sets up a false dichotomy when combined with mankind's institutionalization in conceptualizing - based on general characteristics, creating points of reference from which to make choices based on those perceived institutionalized characteristics; i.e. is one a Republican (R) or Democrat (D). The details of a given issue will be ignored, with the given affiliation to be a representative of the implied position. This false dichotomy is wrong on multiple fronts: with the institutionalization, the R or D come to represent constellations of whole categories of belief, but individuals are in those parties, and those individuals may have ideas which are 'crossing the aisle' and not representative of the perceived R or D; the nature of those serving in government who want to 'fix' things, is a power-hunger who wants to use the legal force of the government to make, or 'guide,' society to behavior that this one in a position of authority thinks they should live; when the two parties are both filled with those who want to fix society, then the choice to people is to which degree control should be placed upon them - a question of degree; however, still has the principle set that government should dictate over the people for their own good.
The system we have was not wanted originally. Two parties set up the false dichotomy. There should be no parties, or many parties for the more parties there would be, the more chances there would exist to follow what those who wrote the constitution intended: a check and balance among the parties from them sniping each other, with the necessary increased complexity of adding extra components - more parties.