<p>What are the down sides of using a composite/compound primary key?</p>

Need more specificity.
Taken too far, it can overcomplicate Inserts (Every key MUST exist) and documentation and your joined reads could be suspect if incomplete.
Sometimes it can indicate a flawed data model (is a composite key REALLY what's described by the data?)
I don't believe there is a performance cost...it just can go really wrong really easily.