Yes, it is absolutely possible. Due to the way different effects are constructed, the exact process varies a little form one type of effect to another, but this knowledgebase article
explains how to reposition each type.
If you still have trouble, other things to look for are masks or garbage mattes which are affecting the shape, or some filters such as the angle blur filter, which may need to be adjusted to match the new position of the effect. If it involves a Gleam Composite the same holds true, the position of the Gleam will likely need adjusted as well, based on the new position of the effect.