We've only seen this in a few limited cases, and have implemented fixes for them in the past.
Can you log into the back-end area and go to "Help->System Info". Then click into the "PHP Settings" tab. Look for the section with "Curl" and report back if you have something different than the below:
curl
cURL support enabled
cURL Information libcurl/7.12.1 OpenSSL/0.9.7a zlib/1.2.1.2 libidn/0.5.6
If it's disabled, can you look for the setting "allow_url_fopen" and report it's setting? That's our fallback, and if both are disabled, you may have to work with your service provider/technical support to enable one or the other.
Finally I'm assuming fields import correctly?
Let us know what you find!