Absolutely!

In many African countries for example, you have more doctors, Ingenieurs etc living and working in the diaspora than in the country.

There are also millions of people who've earned an university degree in their countries but have never had a "proper" job in life, either because they didn't know anybody who could sponsor them, or the job market had nothing to offer. They'll also travel to the West, mostly illegally, to search for opportunities.

With time, it becomes increasingly unattractive for people to stay in their home countries because the shortage of skilled labour makes life harder, more expensive etc for everyone else, so the state expands, tries to do everything, but obviously this only makes things even worse...