If you have never stretch before you can definitely improve your flexibility. If you have stretch regularly all your life, just like if you have done resistance training all your life, you will begin to lose flexibility, just like you will begin to lose strength, as you get older.
Something to keep in mind is that if you stretch, static stretching, before you weight train your muscle will be weaker. Also, as with everything in life, there are trade offs. The more mobility a joint has the less stability it has as well.
But as in everything, genetics has a lot do with how flexible you are and can become. B.J. Penn was never really that athletic and in great shape but what helped a lot in him becoming a Mundials Champion at a Black Belt level in less than four years was his insane flexibility. When trying to pass his guard it was like he had four arms.