The collection is threadsafe, as in "using it concurrently won't corrupt its state". It doesn't make your "contains then add" operation atomic. It's your code that isn't threadsafe

​ ​ConcurrentBag​ ​ is threadsafe, but your code isn't:

if (!SystemNames.Contains(name))
{
    SystemNames.Add(name);
}

​ ​Contains​ ​ will execute in a thread-safe way, then ​ ​Add​ ​ will also execute in a thread-safe way, but you have no guarantee that an item haven't been added in-between.

For your needs, I recommend using a ​ ​ConcurrentDictionary​ ​ instead. Just ignore the value as you won't need it.

var SystemNames = new ConcurrentDictionary<string, bool>();

Then use the ​ ​TryAdd​ ​ method to do the "if not contains then add" in a single atomic operation:

SystemNames.TryAdd(name, true);