America has proven able to cope with a high level of immigration, both economically and socially, precisely because it manages to transmit all that cultural capital to the newcomers. Yes, let us say the dread word: assimilation. Various European nations have had more difficulty assimilating immigrants either economically or socially.
America could almost certainly take more immigrants than it does now without noticeable economic or social effects; so too could many European countries, if they found better ways of integrating the newcomers. (Though to be sure, this may not be culturally easy, or even possible; America is uniquely a nation of extremely diverse immigrants.) But that doesn't mean they could integrate everyone who wants to come, since everyone who wants to come seems to be . . . practically everyone.
America or Europe could easily be demographically swamped if even a fraction of the world's five billion other citizens headed for the West. The physical infrastructure of buildings and roads is not up to a sudden massive increase in population, much less the government institutions. And there is the danger that admitting too many people from countries lacking the cultural capital of capitalism could overwhelm the local culture's ability to assimilate them, destroying the very prosperity they came to seek.
How many is too many? Well, the foreign-born population of America peaked around 1890 at about 15%. Looking around me, I see that almost no one seems to be speaking Czech, Italian, Polish or Yiddish, or even English with a crusty Irish brogue, so I presume they were all assimilated adequately.
Currently, America's foreign-born population is about 10%. This suggests that America could increase its immigration by 50% without destroying its prosperity machine. It's harder to gauge in European countries, which have no established tradition of absorbing massive immigration ...