What breaking point are you imagining? Our ability to feed people? The US produces enough food to feed the entire world.
Not the entire world a US meat based diet, but yes the entire world.
House people? That's more challenging, I admit, but the people coming here are a fraction of our population.
If conditions deteriorate because of overcrowding, you will have fewer people make the trip. I get that outbreaks of war, political oppression, and maybe disease could push people to make even dangerous futile attempts, but economic migrants will find an equilibrium.
Population growth is falling worldwide. There is no danger of overcrowding beyond the point of the land to sustain life. That could be a concern for a nation like Iceland, but not the US, which is one of the most productive and least populated places on earth.
There is a possibility of deteriorating living conditions, in the short term, but those are challenges I am willing to face.