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I followed all the suggested steps online, such as fixing the normals, removing all the doubles, etc. These Are The objects:

The Target, (the one with the modifier applied) is the text and the grid is The object used for the modifier: Starting meshes

This is the modifier applied to the target. "Plane" is the last image's object (something like a grid), and the 3d view is the result, it seems very strange and buggy, like the boolean modifier didn't work well.

This is the result

Any ideas on how to fix this? I am trying to subdivide the mesh in quads that have all a cube shape, but for example in the outer parts of the mesh, will follow the lines of the mesh. Something like consecutive cuts in two directions.

I know also that I can use remesh modifier, but I think is not a good idea to use it because it takes a long time to process and create a lot of geometry.

I am attempting to create a mesh with lot lines along the x and y axis so that I can separate them using separate by loose parts and get lot of cube similar shaped part of the mesh.

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From the way you've set up the boolean modifier, I gather you're attempting to stamp a grid pattern into the front of the text, like this?

enter image description here

If that's not the result you're after, then could you clarify what you're trying to achieve?

If that is the desired result, then it's hard to say exactly what is causing the result you're getting without poking around in the .blend.
Since you already tried the two most common culprits, The next general fix I'd try is sticking a remesh modifier on the text (above the boolean modifier). This will generate a new mesh resembling input, hopefully correcting any non-manifold geometry.

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I think the problem is the memory. I replicated your error using a "siever" mesh with many holes, while using a mesh with few holes (i used four holes) I had no problem

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