Just as I was about to post my question, I figured it out. So, I am posting it anyways incase someone else has the same problem.

THE PROBLEM:
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I am creating a simple dynamic text box with a scroll bar.
Step 1. Select Text Tool
Step 2. Go to the stage and drag a text area.
Step 3. Drag the UI Component "Scroll Bar" over to the stage and place 'on top of' the text box you just created. The scroll bar should conform to the text box and automatically resize itself.

(See specific instructions here: http://www.macromedia.com/support/fl...ng_text_mx.htm)

Thats where my problem started. My scroll bar did NOT conform.

RESOLUTION:
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Go to "View" and select "Snap Objects" -- The UI ScrollBar will now conform/resize/attach/snap to the text box.

Yup. That simple.