Will a strong acid still dissociate "completely" even in the presence of a stronger acid?
Suppose you have nitric acid (about Ka=20) and perchloric acid (Ka > 100,000).  

$\frac{[A-][H+]}{[HA]} = K_a$

In 1M nitric acid alone, let x be [$\ce{NO3-}$], that is dissociated nitric acid 

$\frac{x^2}{(1-x)} = 20$

x = 0.954M 

the nitric acid would be about 95% dissociated.  

If the solution is 1M nitric and 1M perchloric, with all the perchloric acid dissociating, the nitric acid equilibrium would be:

$\frac{x(1 + x)}{(1-x)} = 20$

x= 0.913

the nitric acid would be about 91% dissociated.

For 2M nitric acid alone:

$\frac{x^2}{(2-x)} = 20$

x= 1.83M

or 92% dissociated. 

For 2M nitric and 2M perchloric:

$\frac{x(2 + x)}{(2-x)} = 20$

x = 1.69M

or 84% dissociated.

So yes it can make a difference and it is more apparent at higher concentration.