Proton anisotropy in velocity space has been generally accepted as a major parameter for exciting electromagnetic ion cyclotron (EMIC) waves. In this study, we estimate the proton anisotropy parameter as defined by the linear resonance theory using data from the Van Allen Probes mission. Our investigation uses the measurements of the inner magnetosphere (L < 6) from January 2013 to February 2018. We find that the proton anisotropy is always clearly limited by an upper bound and it well follows an inverse relationship with the parallel proton b (the ratio of the plasma pressure to the magnetic pressure) within a certain range. This upper bound exists over wide spatial regions, AE conditions, and resonance energies regardless of the presence of EMIC waves. EMIC waves occur when the anisotropy lies below but close to this upper bound within a narrow plasma b range: The lower cutoff b is due to an excessively high anisotropy threshold and the upper cutoff b is possibly due to the predominant role of a faster-growing mirror mode instability. We also find that the anisotropy during the observed EMIC waves is unstable, leading to the linear ion cyclotron instability. This result implies that the upper bound of the anisotropy is due to nonlinear processes.