What am I seeing when I look at the objects in the sky that are supposedly satellites using a telescope?
High altitude aircraft designed to fool people like you!
I was waiting for someone to post that. I'm going to refer to your version of the High-Altitude ISS as "ISS-HA", or the International Space Station - High Altitude just for clarity.
Reasons why high altitude aircraft wouldn't work:
1) Telescope Focus. Have you ever used a camera manual focus before? If not, try this simple experiment. Set the focus to 1 meter. Then try taking a picture of your entire street. What happens? Massive blurriness. Similarly, looking at the ISS-HA 10-20km in the sky that's suppose to be 350km will cause this as well.
2) How could you refuel said high-altitude aircraft without anyone noticing? I think I'd notice a massive plane flying up to the middle of nowhere. You're going to need a massive amount of fuel to maintain the speed the ISS-HA is moving at (See point #4). You'd also be able to look out the window of any random airplane and see that the ISS-HA's much, much larger than it should be if it were in orbit.
3) Triangulation. Given enough observational points, you are able to calculate the rough distance. So if 5-10 people go out and spend 50 bucks each, you could arguably verify this.
4) At minimum, the ISS-HA must be moving at minimum 27,700 kilometers per hour. Have you ever noticed how fast the terrain in the horizon is moving when you look out the side of a moving vehicle? It's not very fast compared to the objects / terrain closer to you. Similarly, for the ISS-HA to have the ILLUSION that it's 350km high @ 27,700 KPH when it's really only 50km up, you have to be moving that much faster. The current top speed of any aircraft, I believe, is the NASA X-43 (Unmanned) at Mach 9.8, which is about 12,000 KPH.
That's a pretty sturdy non-aerodynamic structure to be moving at that clip.
P.S. For you knitpickers, I know that the speed I noted of the ISS-HA varies from (1) to (4). Doesn't really matter what the exact number is. Staying within the atmosphere means 35km-50km, a factor of 7-10x closer than it should really be, so a +/- of 30 km really doesn't make too much of a difference.