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I have 3 simple objects: one cube and two cylinders. For some reason, I cannot get simple boolean operations to work.

Example:

Screenshot 1

One issue seems to be the fact that the union object has the same width as the target object. If I decrease the width of the union object slightly or move the cylinders a little bit out of their perfect position it works. Unfortunately, this is not what I want.

Screenshot 2

Finally, there is another issue. I cannot bevel the combined object, even if I apply the boolean modifiers first.

This is a link to the test file: https://www.dropbox.com/t/0YCbtRBy4TgEtHA4

or here:

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Boolean doesn't work when faces are overlapping, at least until the recent versions, maybe they will fix that, so you need to find another method for the union one, maybe simply do it manually.

enter image description here

About the Bevel modifier, you've choosen None mode, so it will bevel every edge, and you've enabled Clamp Overlap option, that won't allow you to do any overlapping. With the given topology you won't be able to bevel anything without overlapping. So what you should choose is the Angle mode, or perhaps Weight if you want various bevel strengths depending on the edges, and disable Clamp Overlap. But even with those options I'm not sure it will be easy to manage your topology.

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