Yes ... It means those in-between years before the teens begin.
Is Ten Years Old A Tweenager?
Some say yes, others say no. This means that the definition isn't clear cut so you can think whatever you want. And if you think ten years old is a tweenager then there's no one stopping you.
According to the Urban Dictionary:
Used to describe (usually in relation to consumption and marketing) a person who has entered the "in between" years before adolescence (see teenager); children between the ages of 10 and 12. These children display common traits, interests, and developing psychologies separate from those in younger, and older age brackets. Commonly abbreviated to "tween".
I personally believe it depends on the ten year old. Some mature faster or slower than others.