For Estes Park and RMNP, June is historically a drier month, in comparison to other months. Showers in May give way to a transition in June, with the monsoons coming in during late July/early August, and sometimes lasting until mid-September.
There have been 3 historical floods in the region, 2 described as a "500 year flood". The first Big Thompson Flood was in July 1976. The 2nd Big Thompson Flood was in September 2013. And there was the Lawn Lake Dam Break Flood in July 1982. These were anomalies and not a part of what one could consider a typical weather pattern for the region.
Whenever one travels in the mountains, whether in Colorado or elsewhere, there's always going to be a chance for adverse weather, sometimes extreme. It's like taking a cruise or visiting islands in the Caribbean during September or October...great time of year for it, but taking a chance on weather/hurricanes.