What you ask is certainly possible, and in (at least!) two different ways. First, you can use GREP to search for this
bul+ ul+b
and manually apply bold (or put “Bold” in the Change To formatting field, and use any of the Change/Change Next/Change All buttons).
Second, a way faster method, by adding a GREP style to your paragraph style. You can use the same GREP for this; all names will automatically be bolded.
If you may have three or more names (“David Lee Roth” comes to mind :D), use this GREP:
bul+( ul+)+b
There is a possible problem, though. It is right there in your example! “Singer Mick Jagger” — that's three capitalized words in a row, but the “Singer” is not part of a name. If you manually apply your bold, you can always adjust it later on, but when you want to use a GREP style you're pretty much stuck to it. Depending on your text, you could try this
(?<=.)bul+( ul+)+b
which will ignore the very first word, even if it starts with a capital. You can even tell GREP to ignore the specific word “Singer”:
(?!Singer)bul+( ul+)+b
.. but perhaps you also have a “Singer/songwriter”, a “Musician”, “Guitar Hero”, or “Ex Genesis Drummer and Singer” ..