A "brute force" method of solving GIS problems that involve acquiring values from raster cells and storing them on spatially overlapping vector features is to first convert the raster to a vector. The main issue with this approach is that converting a raster to vector can lead to very large file sizes - and slow operations. Furthermore, if using ArcGIS without spatial analyst extension, raster to vector converstion is not available.
Could you provide a few of your own examples where you had a project and choose a work-path that purposely avoided a raster-vector conversions? This might have happened either because
i) you had to avoid the conversion because you did not have access to a raster to vector conversion tool, or
ii) you chose to avoid the conversion becuase doing so was a more elegant solution.
EDIT: (Here is an example I used in a previous version of my question before the comments below. It is an example I gave that is referred to in some answers).
Say you had a river network, and you wanted to store the elevation in each line segment in a new column - but elevation values in your GIS database is in raster format (a very large raster). What would be a workaround?