Good questions and having various perspective. The depth means the accurateness of data (what I consider). The only limitation is that how to measure the accurate value for any variable. Since it is not possible to ascertain therefore we are using the term precise (average measurement). So depth of measurement should be a theoretical concept.
The word "depth" is not used in research books, instead the used terms are: Validity, reliability, sensitivity, accuracy and all included under the term: psychometrics of the measurement.
Depends on research paradigm; agree with the two answers above, in quantitative terms; however, in qualitative research, the 'depth' of 'measurement' is how far into respondents' 'life-worlds' one intends to go; from this point-of-view, I think, there is no definite limitation.