Can in any case the faster step of the reaction be rate determining?
I can provide you an example.
The oxidation of formate ion by peroxydisulfate in water solution:
$\ce{HCOO^{-} +  S_2O_8^{2-} -&gt; CO_2 + 2SO_4^{2-} + H^+}$
has the following mechanism:

As you can see, the first step is the slowest, but by using the rate-determining step approximation you wouldn't arrive at the correct rate law which is: $r=k[\ce{HCOO^-}]^{1/2}[\ce{S_2O_8^{2-}}]$. The  first  reaction  is  very  slow,  so  most  of  peroxydisulfate  is consumed  in the third reaction. The correct rate law can be obtained by applying the steady-state approximation to the two radical species.
(This isn't the true mechanism but it is good for showing the point. A  more complex  mechanism  includes  the  formation  of  OH  radicals  and  several  chain termination reactions. That’s why the given rate law is valid only for a limited range of reactant concentrations.)