Henning, follow the advice about putting a bevel or chamfer on the end of your metal stock, and about starting with the proper side of the die.
If you are using a good die, it should have a split in the circumference. It may help to leave the die in your die-stock slightly loose, so that split can spread a bit. Once it is started, you can back off the die and tighten the die-stock screws to get to the proper thread depth on your work piece.
If you are using a die nut (thread chaser) instead of a normal die, they can be hard to start, and they are also usually made of carbon steel, and often not very sharp. Getting a proper adjustable die is the only way to help that situation.
At the diameter you have your piece right now, (6.22mm) it should be quite easy to thread. However, what Lew said about quality dies could be your problem. I've come across cheap far east dies that would not put a thread on undersized brass, much less steel of the correct diameter. A good die will thread that piece at full proper diameter with little trouble.
Dean