Existing works use power correlation matrix to quantify the correlation between Nakagami-m fading channels. However, the power correlation matrix cannot uniquely determine the joint probability density function of the branch amplitudes and the system performance. In this work, the correlation between the Nakagami-m random variables is defined using the correlation matrix of the accompanying Gaussian random variables, and thus the joint probability density function of the Nakagami-m variables can be uniquely determined. We propose closed-form and asymptotically tight upper and lower bounds for error rates and outage probabilities of maximum-ratio combining and selection combining over arbitrarily correlated Nakagami-m fading channels. Analytical results and Monte Carlo simulation illustrate that power correlation of Nakagami-m channels cannot uniquely determine the performance of diversity receptions over Nakagami-m fading channels.