I will have to assume that they already have some DBA's on staff. When I brought up Jr DBA's, this is how I determined when they were ready to advance.
Now I am going to start this off assuming we are talking about a production DBA. Development DBA's are a different breed and for that just do your best on whatever projects they give you. Don't argue (a lot, do express your opinion) if someone more senior says that it should be different than you think.
First of all. You expect them to break something, so you are VERY cautious to give them as little rope as possible. The last thing I wanted was my helping them to make my life a mess.
Don't LIE about what you do. If you break something, tell your senior people as soon as possible to get it fixed.
Don't do or try anything you have been asked NOT to do (this will prevent the fear of telling the truth).
As questions, but be also be cognizant of anxiety. If they are in a panic, don't add to it by bothering your senior people. Just watch and LEARN!
I would expect that most mistakes won't be repeated and NEVER repeated more than once. Remember just like a Doctor "First DO NO HARM!" Should be your rule of thumb for every production DBA.
When in a critical situation (it will happen). DON'T PANIC!!!! Even when you are 100% sure about your course of action.. STOP! Think about everything it will impact again. What you will do if it doesn't fix the problem. Next think about how you might undo what you did. And lastly. Try to take notes some how of exactly everything you are doing. This could be very important to review after the fact in the event you need to do a post mortem report.
If your advisors are anything like me. I simply would grant them more and more responsibilty over time. The more mistakes the longer I would wait until I gave more responsiblity. Understand that I would often stretch new hires out over a YEAR before I let them take a night shift on a production server.
Just to give you an idea of how long it takes. I am a well seasoned production(15 years) DBA and I was a System Admin for 5 years before that. I recently changed jobs and took over an existing site. It took me a good 4 months before I was comfortable with the process at my new company.