There are 4 major spacecraft that can supply the ISS. Progress, Dragon, ATV, and Cygnus. Of these 4 spacecraft, only the ATV has not had an issue in the last 8 months. ATV has stated they will not launch any further vehicles. Bottom line is, every spacecraft that resupplies the ISS has had an issue such that it seems like they just can't launch at this moment. Will this cause problems getting supplies to the ISS?
You forgot HTV, the Japanese vehicle that has a fairly large payload.
It has a planned flight Aug 15, 2015. There are several more HTV flights planned on the manifest.
Progress is due to fly again with cargo in early July 2015. Assuming the Russian space agency is able to resolve its issues that will help with the supply situation.
Cygnus flying on Atlas V is due to fly in December of 2015.
Assuming SpaceX can resolve their flight issue they had flights booked for September and December 2015.
There are issues, but they should be able to work out. With 4 providers (Cygnus, Dragon, HTV, Progress) it should be assumed at least one will succeed. Now would be a great time for ESA to spin back up ATV production and come in as the white knight and save the day! (I know how hard it would be, long lead items, etc but they are putatively developing the the Service Module for Orion based on the ATV already anyway.)