Means the opposite of "broad shouldered" (to act strong and confident and carry alot of burden/responsility in your stride).
People (usually a manager) acquire this title by dodging
work and/or making decisions and generally pass the buck onto a subordinate. This is done to avoid blame or to avoid doing a task that
may be boring.
Officeworker 1: Dave the
manager has just given me a pile of accounts to work on thats going to be externally audited!!
Officeworker
2: But he's more qualified than you to do it! Jeez that
guy has got such slopey shoulders!