The answer to your question depends on the strength of the aluminum, but also on more than just the distance from the edge.
To simplify it, imagine your bronze bushing (and you should tell us if it is Oilite or solid bronze) pressed into another bushing. If both bushings were steel, and had the same wall thickness, and the interference (amount by which the OD of the inner bushing is bigger than the ID of the outer) is 1 part in a thousand (that would be .001" on a 1" bushing), then the inner bushing will compress by about .0005" and the outer stretch by about .0005" Since the elastic modulus of steel is about 30,000,000 psi/in/in, the stress in both parts will be about 15,000 PSI, compression in the inner bushing and tension in the outer. These stress levels would be acceptable for almost any steel under almost any conditions. This very simplified example ignores loss of interference by mashing-down (a technical term) of surface roughness of both parts.
You can look up the elastic moduli of your materials. .0003" interference on a 1/4" OD bushing is all you want, but I do not know what facilities you have for making a hole to accurate size, nor for measuring. If the wall of your part is thin on one side and much thicker on the other, then the thicker will support more of the spreading force, relieving some of the stress on the thinner, but I would recommend NOT making the thinnest wall any thinner than would be needed to support the stress from insertion of oversized bushing in another bushing of uniform wall, PLUS the service stress from whatever your machine does.
The more you tell about your application, the better the advice you will get here. One thing that sometimes earns snide and unhelpful replies is a question from someone obviously without much mechanical knowledge that asks for only one tiny detail, as if the questioner knows everything else. As perhaps my reply illustrates, there are no simple questions, and you cannot get a good answer without knowing lots of details about the whole situation. A person who knows little often reveals that by assuming that the question is simple and surrounding details do not matter.