From what I've heard, it's impossible (or rare) for people of Europe's "middle class" to rise into the "upper middle class" or "upper class" the way people do here in the US.
Is that true?

The "class" structure is different in that "rising" is defined differently in other realms at times other than mass financial wealth. However, its just as possible to start a successful business there from the ground up even more so in some industries. The fashion and luxury goods industry for one has made many rags to riches stories in Europe, way more than the United States.
It depends on many factors as it may be more difficult due to a better educated society as a whole as well as a more financially stable population with less economic disparity in Europe than the United States.
However, the premise of being impossible or rare is not true whatsoever.