I really have my doubts about this particular technique. If we are talking about striking the 'brachial plexus' it's a large nerve cluster covering most of the neck and shoulder that connects the arm muscles and nerves to the spine.
However several things to note. It's a pretty big nerve cluster, but most of it is under the muscles on the neck. Hitting muscles is usually not a good tactic in martial arts because the pain this causes is fairly superficial. Hitting the side of the neck seems unlikely to do anything much for this reason.
Shorinji Kempo (a martial art with something of a speciality in nerve attacks) does not even list this as a target. They have 138 distinct targets, many of these are on the neck but these muscles on the side of the neck are not considered a target in themselves.
What they do list as possible targets are several points on the back of the spine. Some of these are on parts of the brachial plexus
NB. I really don't think you should go hitting yourself there. The spine is a fragile thing and nerves even more so.
Another part of the neck that really does cause disorientation when hit is the carotid artery. My guess is that this is usually what is going on when people say they are doing a 'brachial stun'.
The internal carotid artery can be damaged, but this is very dangerous and requires pushing your finger an inch or two into the soft part of the neck to attack.
Don't do this either, my instructor randomly fainted several times over the course of the next month after this technique was demonstrated on him
The external carotid artery is often used as a target for strangles. I have been taught a method of striking this the vagus nerve on this artery to cause a stun. Though there are some people on this site that contest this techniques efficacy.
Even just touching this point is very uncomfortable. I would usually show my training partners in class where to find it by touching it with my smallest finger. In all honesty I don't know 100% that you can cause a stun by striking it because I haven't ever had a chance to try this. But on the balance of probability it seems not just plausible but likely that this would cause a large effect of some kind.
The bottom line is the neck is a very fragile part of the body. You are lucky you didn't hit anything important.