Yes ... It means those in-between years before the teens begin.
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.