I hadn't heard that about applying twice... I did this, although I used different rec providers and applied to (mostly) different schools. There are other threads on TM about applying twice that come to very different conclusions: if you are rejected from a school, you have the right to ask them to discuss ways you could improve for a second go at it, both there and elsewhere.
More on my situation...
On the first go, I was rejected by: U Washington, U Oregon, UC-Irvine, UCo-Boulder, and Oregon State. Washington State U let me in to their PhD program w/o funding, and I turned this down to attend a U.S. MA that offered funding and tried again, with much better results (chiefly, UNC). UW has the distinction of being the only program that has rejected me twice (although they did waitlist me the second go around.)