"degenerate" usually means a structure that meets the desired technical defintion, but is't particularly useful. For example, a point might be a degenerate circle, but points are not all that useful for studying circles. Likewise, two intersecting lines can be degenerate conics, but, again, they don't tell us much about hyperbolas, ellipses, or parabolas.
Speculation: A degenerate vector in R^n might be a vector in R^n that has too many zeros - maybe any. Perhaps it is trying to tell you that none of the vector values is zero.
Personally, I would want to find the intent of the word in the specific context. If the book doesn't actually define the term, I might be tempted to call the author. Maybe it will get into the next revision.