Starting another instance of a note with the same channel and key as a note that is already sounding is technically allowed in MIDI, but I believe it is not normal.
In particular, the three notes mentioned in the question look like this in a sequencer:
It seems likely the overlapping notes are supposed to be separated and have a similar duration as the surrounding notes. Perhaps a glitch in the MIDI capture process didn't catch the correct release of those notes. (This MIDI file also contains a few notes that are 1 tick in duration -- 1/960 of a second in this MIDI file, which seems unlikely to be how the pianist originally preformed them.)
In MIDI, each message that starts a note (a Note On message for a particular channel and key with non-zero velocity) must eventually be followed by a message that ends the same note (a Note On message for the same channel and key with zero velocity, or a Note Off message for the same channel and key with any velocity).
It is possible to start another instance of a note with the same channel and key as a note that is already sounding, but each instance must eventually have a message that ends the note.
However, when there are overlapping notes with the same channel and key, it is ambiguous which end goes with which start. For example, if you encounter the following sequence of messages on the same channel at beat intervals:
start C5, start C5, end C5, end C5.
The sequence could be interpreted two ways:
start a, start b, end b, end a.
start a, start b, end a, end b.
Displayed another way, imagine the following two staves are two voices on the same track and channel. The first and second measures show the two interpretations above.
Because of this ambiguity, I believe overlapping notes of the same channel and key are usually avoided in MIDI.
Here is what the MIDI spec has to say about this. From the MIDI 1.0 Detailed Specification document version 4.2.1, PDF page 67 (printed page A-4):
ASSIGNMENT OF NOTE ON/OFF COMMANDS
If an instrument receives two or more Note On messages with the same key number and MIDI channel, it must make a determination of how to handle the additional Note Ons. It is up to the receiver as to whether the same voice or another voice will be sounded, or if the messages will be ignored. The transmitter, however, must send a corresponding Note Off message for every Note On sent. If the transmitter were to send only one Note Off message, and if the receiver in fact assigned the two Note On messages to different voices, then one note would linger. Since there is no harm or negative side effect in sending redundant Note Off messages this is the recommended practice.