“Higher” than what?
If you mean higher than Harvard’s rate (< 10%), the answer is, “of course”.
But even if you mean “higher than 50%”, the answer is the same.
Most research universities have acceptance rates higher than that.
There are about 3000 4 year colleges and universities in the USA.
https://nces.ed.gov/fastfacts/display.asp?id=84
Of these, about 300 are classified as research universities in the Carnegie classification system
(doctoral/research universities,“high research activity” universities, or “very high research activity” universities).
https://en.wikipedia.org/wiki/List_of_research_universities_in_the_United_States
And that does not even count the many universities that grant Masters degrees, but few or no doctorates
(like the ones that show up in the US News lists of “regional” universities).
Only a fraction of those ~300 research universities (maybe 50-75) have admission rates below 50%.
For example, Indiana University was tied for #76 in the US News national research university 2014 ranking.
In 2013, its admission rate was about 72%.
Wichita State is a “high research university” for which US News does not report a rank.
In 2013, its admission rate was 95.8%.