Dimensions and notes moving after opening drawing.

Are the dimensions brought in from the model, or inserted in the Drawing with the Smart Dimension tool?

You can fix dimension & note position. Select note (dimension), there is an option to fix position.

Depending on your scenario, there are one or several document and system settings that may apply to the behavior you are seeing.   Try setting these:

• System Options>Drawings>Reduce spacing when dimensions are deleted or edited (add or change tolerance, text etc..) - Uncheck This setting allows SOLIDWORKS to keep your dimensions offset to distances defined in your Document Properties. This is a live setting that will always apply changes to dimensions when SOLIDWORKS detects dimensions do not follow offset rules.  This is the most likely culprit to the behavior you see.
• Document Properties>Dimensions>Annotation view layout - Uncheck  This setting applies your placement of dimensions in the model to your drawing.  This is a very unlikely culprit to the behavior you see, unless you are making extensive use of model items.
• Document Properties>Dimensions>Center between extension lines - Uncheck  This setting allows SOLIDWORKS to make sure your dimension text for linear dimensions is always centered between the extension lines.  This is a possible culprit for the behavior you see.

These are the automated dimension trinity of settings that you turn off when you want to fully control of dimension placement instead of entrusting SOLIDWORKS to help you.  These settings are for some people, but obviously not all.

Edit: HOWEVER, these settings do not apply to Notes.  Nor do these settings cause dimensions to move to wild locations on the drawings (quite the opposite, actually).  If you are seeing behavior where dimensions and notes are shifting about wildly, then this is something that needs to be looked into with your reseller.  In the meantime, you can use the above settings to see if they affect the results you see.  System Options are system level settings, so it's a set once and forget.  However, Document Properties need to be set in your Drawing Templates for future benefit, and within each existing (already created) drawing for benefit within that document.

Adam, this sounds like very strange problem.  It is also a major pain. 

I thought I could do something like recreate the problem but this is not going well.  I have seen in in years past and it was usually after editing the part that some of the dims would go nuts, and balloons and notes attached to an edge would not be where they where supposed to be. 

Just to clarify, are you saying that a drawing will look fine, you save and close it, then when you open it again some annotations have moved, even if the model wasn't edited in any way?