If you HAVE Cherokee heritage...you should be proud.
Unfortunately, there is a real phenomenon in this country of people falsely or erroneously claiming to be "part-Cherokee"...for example, it is a VERY easy thing to say you are 1/8 Cherokee in 2011...you WOULD have tribal enrollment in either Cherokee Nation of Oklahoma or Eastern Band (Qualla Rez).
If you are not enrolled...then you have a Myth. No doubt about it.
Trail of Tears was 1838/9....the majority of the tribe removed to Indian Territory at this time...only a small number of full blood/traditionalists remained in the mountains to coalesce and become the Eastern Band. There were also a number of mixed bloods, most of them intermarried with non-Cherokee spouses...that CHOSE to stay in the east. They are found on numerous Cherokee "East of the Mississippi" Rolls. If you descend from these people (who were scattered in various locations and were not part of any Cherokee community...your blood quantum would be QUITE low (and most certainly NOT 1/8).
There were NO Cherokee communities left intact after late 1830's...early 1840s OUTSIDE OF Oklahoma or Quallla Boundary area that could produce full blood or heavy blooded Cherokees. Most of the people who claim a "full blood" Cherokee great-grandparent refuse to believe this historical reality. They cling to the myth. And in many cases, people are hearing a story of Cherokee blood that has morphed and changed with the retelling...so that it reaches the descendant as "so-and-so was a Cherokee Indian." Perhaps the original information was radically different: GGGGrampa won a Cherokee Land Lottery (i.e. took Cherokee land)....GGGgrampa was a trader in Cherokee Nation or grew up with Cherokees (i.e. original claim didn't claim Cherokee blood)...or...GGG-Gramma claimed "Cherokee blood" (but, she was a light skinned african american who was "passing for white" and claimed Indian blood to help explain dark features)...there are lots of reasons for hearing these stories of "cherokee blood"...like the game of Telephone...the original message gets changed with the retelling...so that when it comes down to the modern descendant it is RADICALLY different.
By definition...these stories remain a MYTH if you do not have PROOF.
If you cannot point to a TRIBALLY AFFILIATED and RECOGNIZED Cherokee ancestor, you cannot claim this heritage. Period. To claim "hidden heritage" or make a claim that counters historical movements and the reality of Cherokee history is to follow a Myth.