Nice chart. Yes, in theory, any woodwind should be able to do four octaves. Well in theory at least - playing it might be a TAD hard. I give up at around C7, mostly because I just don't usually play music that has notes up that high, preferring rock and jazz much more lately.
Listen to the Cuban salsa/Latin jazz flute players that play F4 with ease. They had to battle trumpets, and are incredible! The C4 is in the normal range, not screechy, but to be played with a beautiful tone. Don't attempt the altissimo register until you can do this!
I once hit a 4th octave E...I was talking on the phone when I hit it.
I'm much more impressed that you can talk on the phone and play the flute at the same time than I am that you can hit high notes.
Let me hijack this thread a bit momentarily and change the question slightly. How do you or would you react to pieces that involve a high Db? (That's the Db three octaves and a half-step above middle C.)
Judging by your fingering chart, I think you're an octave off. Your chart goes to the D three octaves and a major second above middle C. This is the standard three-octave normal usable range given by most orchestration books plus one whole step. If we call middle C C4, as the other members have been, then our high C as shown on the chart is C7.