I noticed that you are wondering what will happen if you purchase a Slingbox PRO-HD and your Internet connection keeps dropping. I will recommend you to contact your Internet Service Provider, since they can troubleshoot and explain to you why this happens. Regarding the Slingbox's performance, the faster Internet speed you have at home (upload), better quality you will get. With that being said, you might receive choppy video or video quality issues if your Internet speed does not meet the requirements for high definition on a Slingbox PRO-HD.
Feel free to check the following link for additional information:
I hope this clarify your questions.