Upvote:4
You may not be present in the Schengen area before the "from" date of your visa nor after its "to" date. I assume that you actually meant to ask about a trip from 6 to 16 June, since the period from 6 to 16 May is in the recent past.
If you enter on 6 June, or on any day between 27 May and 10 June, you will have to leave the Schengen area on or before 10 June. So the answer to your question is no, it is not okay, and yes, you have to travel between 25-05 to 10-06-2019.