Case 1:
Justin Timberlake's "Mirrors", Ariana Grande's "The Way", and Lorde's "Royals" all charted on Urban radio yet weren't allowed to chart on the R&B / Hip Hop chart (yet Lorde's "Royals" became the top "Rock" song of 2013.

).
Coldplay's "A Sky Full Of Stars" is now the #1 song on Billboard's Hot Rock Songs.
I know people will also complain about "We Are Never Ever Getting Back Together" & "Cruise"'s runs on Hot Country Songs, but they at least had an original country version or another country version available even though for "Cruise", it was long done on country when it spent 20+ weeks at #1 on the chart thanks to the Nelly pop remix.
They really need to start basing material on the SONG and not just solely what the artist mostly releases.