解:因为$\vert a\vert = 10,$$\vert b\vert = 6,$
所以$a = \pm10,$$b = \pm6。$
分四种情况讨论:
①当$a = 10,$$b = 6$时,$a + b = 10 + 6 = 16;$
②当$a = 10,$$b = -6$时,$a + b = 10 + (-6)=4;$
③当$a = -10,$$b = 6$时,$a + b = (-10)+6=-4;$
④当$a = -10,$$b = -6$时,$a + b = (-10)+(-6)=-16。$
综上所述,$a + b$的值为$\pm16$或$\pm4。$