The aging process begins after about the age of 25. Sarcopenia (muscle loss) begins.....and worsens as we get into our 50's (our 6th decade of life) as our nervous system also starts to down regulate and decrease our ability to display power.
That said, we know that weightlifting, while it can't PREVENT aging from happening, it can DRASTICALLY slow down the process and make things like sacropenia, power decreases, and strength decreases less of an issue. This is why it is so vital for people to lift weights later in life.
As far as making progress goes, you can make progress at any age, it is all about how you train and maniupulate the variables. You may not be as strong as you were when you were 18 or 19 (I know I am not) and you may not be able to recover as quickly from intense training bouts, but that doesn't mean you can't progress and increase strength/muscle mass, increase power, increase flexibility/mobility and enhance work capacity.
If you want to hear about progress later in life......I had a 56 year old man who was HIV Positive (muscle wasting away!) squat 335 for 5.