I, as a student of life and culture, have one answer for you, and it makes sense, based on a number of sociology and anthropology books Ive read.
The Native Americans do not want any more of their culture and beliefs taken, twisted, manipulated or otherwise hijacked by outsiders. The Dreamcatcher, like many other items & practices, are sacred to their culture and their beliefs. While as non-Natives, we may admire and/or recognize the value these things have, I dont think its appropriate to incorporate them into other practices.
A similar issue is for non Natives to attend sweat lodges. While the process itself is physically very intense, and therefore easily not for everyone, it is also a spiritual practice, with months and months of preparation to enter a deep meditative state.
It is sad- but, nonNatives, and us Yanks in particular, have taken so much from the Native Americans at our own discretion- land, natural resources... life! I don't think Americans know that there were over 25 MILLION Native Americans who perished- through the course of Columbus' 'discovery' of the 'new world'. Through smallpox, internment, relocation, destruction of food and subjegation- I think that 'we' owe them more than we can ever give. (my personal opinion- which every christian should also accept as a legacy sin- just like 'original sin' something we inheritedf from generations' past)
We should neither stand in awe of nor have disdain for these peoples. But we should respect their culture as their own. That is not to say that we must ignore or isolate "them", but our engagement with "them" must be as individuals, with the same respect we want for ourselves.