In the intelligent reflecting surface (IRS) assisted communication systems, the acquisition of channel state information (CSI) is a crucial impediment for achieving the passive beamforming gain of IRS because of the considerable overhead required for channel estimation. Specifically, under the current beamforming design for IRS-assisted communications, $KMN+KM$ channel coefficients should be estimated if the passive IRS cannot estimate its channels with the base station (BS) and users due to its lack of radio frequency (RF) chains, where $K$, $N$ and $M$ denote the number of users, reflecting elements of the IRS, and antennas at the BS, respectively. This number can be extremely large in practice considering the current trend of massive MIMO (multiple-input multiple-output), i.e., a large $M$, and massive connectivity, i.e., a large $K$. To accurately estimate such a large number of channel coefficients within a short time interval, we devote our endeavour in this paper to investigating the efficient pilot-based channel estimation method in IRS-assisted uplink communications. Building upon the observation that the IRS reflects the signals from all the users to the BS via the same channels, we analytically verify that a time duration consisting of $K+N+\max(K-1,\lceil (K-1)N/M \rceil)$ pilot symbols is sufficient for the BS to perfectly recover all the $KMN+KM$ channel coefficients in the case without noise. In contrast to the conventional uplink communications without IRS in which the minimum pilot sequence length for channel estimation is independent with the number of receive antennas, our study reveals the significant role of massive MIMO in reducing the channel training time for IRS-assisted communication systems.