#Mathematica 75 69 bytes
Loquacious as usual. But Martin B was able to reduce the code by 6 bytes.
f@a_:=!Or@@Flatten[a~Drop~{#}~StringStartsQ~a[[#]]&/@Range@Length@a] f@{"111", "010", "000", "1101", "1010", "1000", "0111", "0010", "1011", "0110", "11001", "00110", "10011", "11000", "00111", "10010"} True
f@{"He", "said", "Hello"} False