I believe I know what's happening to you. I was experiencing the same behavior in my PC using Arduino 1.6.6. In my case I was using a VPN that was causing the issue. The moment I exited the VPN my IDE was able to download the i586 add-on. Do you think something like this could be happening to you? Maybe you are not on a VPN but a firewall could be causing the same effect, is there any restrictions in your network?
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Do you have updates on this? Have you been able to download the Board packages on your PC?
Have you tried deleting the Arduino15 folder in C:\Users\you\AppData\Roaming for 1.6.5 and C:\Users\you\AppData\Local for 1.6.6?
Thanks for the answer. I think the firewall of the company has blocked the Arduino ID. I also have the Arduino board at home tested. There I was able to install it properly. Now i will try to turn off the firewall briefly.
Thanks for your answer. I think Peter is right with the network blocked. But I will also test to delete the arduino folder you say.