Solution:

The sum $\dfrac12 + \dfrac13 + \dfrac17 + \dfrac1n$ is greater than $0$ and less than $\dfrac12 + \dfrac13 + \dfrac17 + 1 \lt 2,$ so as an integer it must equal $1.$

Since $\dfrac12 + \dfrac13 + \dfrac17 = \dfrac{41}{42},$ we need $\dfrac1n = \dfrac{1}{42},$ so $n = 42.$

Then $2,$ $3,$ $6,$ and $7$ all divide $42,$ but $n = 42$ is not greater than $84.$ So the false statement is $n \gt 84.$

Thus, the correct answer is E.