I'm really bad at explaining things in detail, but I'll give it a go (I made a typo in my post; I meant Storytelling.)
The thing that makes that kind of system less than ideal is that it is too rules heavy for a narrative-focused game, in my opinion; if you think about it, Attribute+Skills is actually quite rigid, and it does lead to slowdown when you get to certain points. For example, if the system has a lot of Attributes and Skills, you run the risk of "should I use Speaking for this one, or Presence?"; again, in my opinion, if you need to stop to ask yourself which one is best suited, then you are slowing down the game, which eats up precious time.
On the other hand, if you have a very low number of Skills and Attributes, you run the risk of having a single small set of them becoming very, very important.
On the other hand, take a look at Wushu, or (the example that will probably be more familiar) Fate/Risus. There, you pick whichever trait the character has that would be most appropriate to that particular situation; this also has the benefit of role protection, because by simply writing Doctor on your sheet, you are better than someone who did not write Doctor on their sheet.
If you want a better explanation, PM Xefas about it, and he'll give you the low-down.