We present line profiles and profile parameters for the Narrow-Line Regions (NLRs) of six Seyfert 1 galaxies with high-ionization lines: MCG 8-11-11, Mrk 79, Mrk 704, Mrk 841, NGC 4151, and NGC 5548. The sample was chosen primarily with the goal of obtaining high-quality [Fe VII] lambda6087 and, when possible, [Fe X] lambda6374 profiles to determine if these lines are more likely formed in a physically distinct 'coronal line region' or are formed throughout the NLR along with lines of lower critical density (n(sub cr)) and/or Ionization Potential (IP). We discuss correlations of velocity shift and width with n(sub cr) and IP. In some objects, lines of high IP and/or n(sub cr) are systematically broader than those of low IP/n(sub cr). Of particular interest, however, are objects that show no correlations of line width with either IP or n(sub cr). In these objects, lines of high and low IP/n(sub cr), are remarkably similar, which is difficult to reconcile with the classical picture of the NLR, in which lines of high and low IP/n(sub cr) are formed in physically distinct regions. We argue for similar spatial extents for the flux in lines with similar profiles. Here, as well as in a modeling-oriented companion paper, we develop further an idea suggested by Moore & Cohen that objects that do and do not show line width correlations with IP/n(sub cr) can both be explained in terms of a single NLR model with only a small difference in the cloud column density distinguishing the two types of object. Overall, our objects do not show correlations between the Full Width at Half-Maximum (FWHM) and IP and/or n(sub cr). The width must be defined by a parameter that is sensitive to extended profile wings in order for the correlations to result. We present models in which FWHM correlations with IP and/or n(sub cr) result only after simulating the lower spectral resolution used in previous observational studies. The models that simulate the higher spectral resolution of our observational study produce line width correlations only if the width is defined by a parameter that is more sensitive to extended profile wings than is the FWHM. Our sample of six objects is in effect augmented by incorporating the larger sample (16 objects) of Veilleux into some of our discussion. This paper focuses on new interpretations of NLR emission-line spectra and line profiles that stem directly from the observations. Paper 2 focuses on modeling and complements this paper by illustrating explicitly the effects that spatial variations in electron density, ionization parameter, and column density have on model profiles. By comparing model profiles with the observed profiles presented here, as well as with those presented by Veilleux, Paper 2 yields insight into how the electron density, ionization parameter, and column density likely vary throughout the NLR.