Is the result of adding several positive semidefinite matrices also positive semidefinite?

I have a certain number of nxn matrices that are positive semidefinite. Is the result of adding all these matrices also positive semidefinite? If affirmative, is it always the case or, instead, in just some instances?

I would also appreciate if you can provide me with the proof or give me a reference I can consult.

$\rm\bf Observation$: $\hskip 1.5in x^*(A+B)\,x=x^*A\,x+x^*B\,x$
$\rm\bf Rhetorical\;\;Question$: Now what happens if both $x^*A\,x\ge0$ and $x^*B\,x\ge0$?