Each year there is a period of about a week when the I.S.S. remains in the sunlight and is visible on every pass from the northern hemisphere. The reason is that it's orbit aligns pretty well with the angle of the Earth's shadow. It's easier to show it than it is to explain it.
Here's a photo in map mode which shows the orbital path and it's relation to the shadow.
Being a still photo, it doesn't show how the rotation of the Earth works with the orbit's precession, so I made two videos to illustrate it.
The first video shows a night's worth of motion. The view shows the United States with the equator being roughly across the middle of the globe. It starts at dusk on July 7, 2009. Each step is 5 minutes in time.
Shadow video 1
This video shows the orbit from above the North Pole. The I.S.S. can't be seen from there, it doesn't come far enough north. The video shows the orbital path and it's relationship to the Earth's shadow. These videos are a little strange to watch because the Earth appears to stand still and the shadow moves. Actually, the Earth is spinning and the shadow is pretty much stationary.
Shadow video 2
Videos were made with EOO. I simply ran the program and pointed a video camera at the screen.
The same thing happens in the southern hemisphere, during their summer. During winter the angle of the Earth's shadow causes the ISS to be sunlit in dark skies only shortly before sunrise or shortly after sunset.
Here's a photo like that above, except in the winter. You can see the different angle of the shadow.