That could be the case, but it is a fairly common lab reaction of t-butyl halides go by first order kinetics.
Only in the presence of weak nucleophiles.
The reaction is performed with NaOH. The reaction goes by first order kinetics It isn't the pKa of the base, but the concentration.
Undoubtedly, it can go by second order kinetics if base concentration is increased.
Base strength, not concentration, causes the shift from unimolecular to bimolecular reactions.
At low concentrations, the solvent causes first order bond cleavage faster than a second order E2 elimination. If the base concentration is increased, at some point, the second order reaction become predominant.
This can be the case for many second order reactions in protic solvents. Since you know t-butyl bromide has first order kinetics with dilute NaOH, the NaOH is simply not showing its presence in the reaction kinetics. If the concentration is doubled, the reaction rate doubles. If the NaOH concentration is continuously increased, at some point it will become predominant. You can think of it like this. Any of those secondary or tertiary halides should have second order kinetics. If the reaction is diluted, at some point the solvent effect can become greater than the nucleophile/base. This should be consistent with reaction kinetics. Second order reactions only take place by the molecular collision. The number of collisions depends upon concentration. At some point, the solvent can compete with the base/nucleophile.