We study the problem of time series imputation in multivariate neural recordings. Compared to standard time series imputation settings, new challenges for imputing neural recordings include the lack of adjacent timestamps for electrodes missing over days, and generalization across days and participants with different electrode configurations. Due to these challenges, the standard practice in neuroscience is to discard electrodes with missing data, even if only a part of the recording is corrupted, significantly reducing the already limited and difficult-to-obtain data. In this paper, we establish Deep Neural Imputation (DNI), a framework to recover missing electrode recordings by learning across sessions, spatial locations, and participants. We first instantiate DNI with natural linear baselines, then develop encoder-decoder approaches based on masked electrode modeling. We evaluate DNI on 12 multielectrode, human neural datasets with naturalistic behavior. We demonstrate DNI's data imputation ability across a broad range of metrics as well as integrate DNI into an existing neural data analysis pipeline.