In this paper, the performance of a distributed intelligent reflective surface (IRS)-aided communication system is investigated. To this end, the optimal signal-to-noise ratio (SNR) achievable through the direct and reflected channels is quantified by controlling the phase-shifts of the distributed IRS. This optimal SNR is statistically characterized by deriving tight approximations to the exact probability density function and cumulative distribution function for Nakagami-$m$ fading. Thereby, the outage probability and achievable rate bounds are derived in closed-form, and they are validated via Monte-Carlo simulations. Our numerical results reveal that the distributed IRS-aided communication set-ups can boost the outage and rate performance of wireless systems.