In addition to the technical issues when an aspect ratio which is between 4:3 and 16:9 is rendered as 16:9 and then shown on a 4:3 set, another reason some older movies have borders on all four sides is that while historical projection practice used to be to project an image slightly smaller than the screen, thus showing the nearly the entire frame of a movie (to allow for imperfections in film advancement, the top and bottom were usually masked slightly in the projector) modern practice often favors projecting all the way to the edges of the white screen, with some of the frame either landing on the black border or missing the screen entirely, and thus being obscured in either case.
Consequently, modern projection routinely crops out part of the picture on the film, and the exact amount will vary depending upon projector adjustment and focusing. A similar thing happens when material is shown on CRT television sets. Because television set calibration has historically varied considerably with temperature, and because it would be irksome to sometimes have significant black borders but sometimes have one or more sides with no border, television sets are adjusted so that even when the image is at its smallest it will reach all the way to the edges of the screen. As a consequence, many television sets will crop out a noticeable amount of the picture.
To deal with uncertainties in projection or display, modern films are shot using what's called a "safe zone", and view finders are marked with two nested rectangles. The outer rectangle indicates what will be captured by the film or video, and directors should keep anything which shouldn't be seen by the audience outside that area. The inner rectangle indicates the part of the frame that audiences are "guaranteed" to see when using any properly-calibrated setup--any theater or television set which would obscure part of the inner rectangle should be deemed to be out of calibration. Directors generally ensure that anything which is interesting should be within the inner rectangle.
Because older films were shot without regard for a "safe zone", it's not uncommon for them to include significant details near the edge of the frame. When mastering such films to DVD, it is necessary to either risk having that material be invisible to some viewers, add borders on the screens of other viewer, or perhaps compromise by adding a small amount of border on all sides of the stored image which would be larger than the amount of image that would get cropped by most television sets, even if some television sets might still crop it a little bit.