I posted a reply over in Kenpotalk, and tought I'd bring it over here as well, hope it helps.
Chen style in particular has been known to contribute to knee problems. The stances are low, and if you aren't sure you are doing them correctly they can get injured. I had a sore patellar ligament in my knee for almost a year because of it. It finally got better, but was a real problem during that time.
In addition, Chen issues Fajing, or release of energy. It manifests in various ways, including stomping the ground. But this isn't simply "stomping the ground". There is more to it than that. If this is done incorrectly it can also cause problems. In addition to joint problems, I have heard that the shock can travel into the head and sort of scramble your brain, over a long period of time.
If you try to do some complex breathing methods while practicing your tai chi, you can also hurt yourself if you are doing it wrong. You don't necessarily need to do complex breathing methods in Tai Chi. Often, natural breathing works very well, and the complex methods that can take you to a higher level should only be done with proper instruction.
Some lineages of Chen, such as Feng Zhi Xiang in Beijing has made changes to the system and softened the stances and the stampings in order to counter these problems. I believe his method is now know as Huan Yin Chen Tai Chi, which I believe is something like Primordial Energy, or something. My understanding of it is far from perfect, but my sifu is a student of Master Feng.
So if you just try to mimick a video, yes it is possible to do something that can hurt you. Perhaps this is less possible with some of the Yang sets that have been simplified and standardized for the masses. But overall, this gets back to video training: it isn't a good idea, your results will be minimal, and you can actually hurt yourself if you are doing it wrong. Not guaranteed, but definitely possible.
If you try to do the complex breathing based on a video instruction or vague description that you have heard somewhere, you will probably get it wrong and that could hurt you.
So the short answer is: Yes, if done incorrectly, Tai Chi can hurt you. Some simplified methods have much less chance of that, but it is always possible to get burned by something you don't understand.