A glimpse out of your window at around 5am today would have answered your question.
Dawn (sunrise) was at 5:14am today. At 5am it was almost broad daylight, it�s just that the sun was not above the horizon. Twilight began at about 3:30am and at just before dawn it was most certainly a lot less dark than it was at 1am � even allowing for the bright moon (waning, with about 65% illuminated).
I think (though I may be wrong) that the phrase is purely used to describe a sequence of events which appear to get worse before getting better.
Karmgirl�s unrelated idea is not so fanciful as it sounds. You can indeed �chase the sun� in an aircraft. Quite how fast you need to go to keep up depends on your distance from the equator. At the equator you would need to travel in excess of 1,000mph to avoid seeing the sun set. At higher latitudes this is much reduced. Concorde used to cross the Atlantic in 3 hours, although the time difference is 5 hours. So if you left London as the sun was setting you would arrive in New York two hours before sunset.
Unfortunately conventional aircraft are not fast enough and a journey to Australia, which takes about 24 hours, sees a time difference of only twelve. You delay the sunset if you travel westwards from the UK, but see it earlier if you travel eastwards.