What is happening is that, in
She is 16 years old.
"old" is an adjective modifying "she", and "16 years" is a noun phrase that is acting as an adverb and modifying "old". You can see this by looking at similar sentences:
The festival was a week long.
Here, "a week" is a noun phrase modifying "long". You can tell that it's a noun because you need to put the article "a" before it. When you put the modifier before "festival", you get "The week-long festival". You have to drop the article, because now "week-long" is an adjective.
In English, I believe (now watch somebody prove me wrong) that adjective phrases of this type before nouns cannot be more complicated than the form number-unit-adjective, where the adjective is restricted to some set of adjectives (deep, thick, long, tall, high, wide, late, early, and so on; I don't know whether this class of adjectives has a name). So you can have phrases like
a 1000-foot-deep lake,
a two-hour-late train,
a 10,000-man-strong army.
When they're not immediately before the nouns they modify, you can have much more complicated phrases.
The lake was a strenuous four-hour hike up the mountain.
Here "a strenuous four-hour hike" is a noun phrase acting as an adverb and modifying the preposition "up". If you try to make this into an adjective coming before "lake", you find it's impossible in English. Certainly, anything like "*the strenuous-four-hour-hike-distant lake" is bad grammar.