Made my own TwitterClient and I think it's better than the example one

And you didn't change a thing? That's strange indeed. I could test your code on my own Arduino and network if you'd like.

Just attach your code to a reply and I'll give it a try. If it works on my end, you might have network issues. OR you might have set the interval to low so your Arduino is asking for updates way to fast and your IP got banned.