Floating point numbers are used. These have rounding errors. To combat that
you should try to make your shapes involved in the difference() operation a
little bit bigger.

// No
difference() {
cube(2);
cube(1.5);
}

translate([5, 0, 0])
// YES
difference() {
cube(2);
translate([-0.1, -0.1, -0.1]) cube(1.5+0.1);
}

And a cube shouldn't take $fn as a parameter.

I hope this solves your problem.

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No, I don't think this is it. These are your problem:
*translate([0,.5,-6]) cylinder(1,d=20, center=true);
*translate([0,1.5,-3]) cylinder(1,d=20, center=true);
*translate([0,1.5,0]) cylinder(1,d=20, center=true);
*translate([0,1.5,3]) cylinder(1,d=20, center=true);
*translate([0,.5,6]) cylinder(1,d=20, center=true);

No idea why.

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c = 0.002;

fn_cyl=40;

difference() {
cube(20, center=true);
union() {
translate([0,-15,0])cube([21,20,21],center=true);
cylinder (21, d=16, center=true, $fn=100);
rotate([90,0,0]) translate([0,0,-6]) cylinder(12, d=16, center=true);

rotate([90,0,0]) translate([7.5,7.5,-11]) cylinder(6, d=2.5,d2=2,

$fn=fn_cyl);
rotate([90,0,0]) translate([-7.5,7.5,-11]) cylinder(6, d1=2.5,d2=2,
$fn=fn_cyl);
rotate([90,0,0]) translate([7.5,-7.5,-11]) cylinder(6, d1=2.5,d2=2,
$fn=fn_cyl);
rotate([90,0,0]) translate([-7.5,-7.5,-11]) cylinder(6, d1=2.5,d2=2,
$fn=fn_cyl);

translate([0,.5,-6]) cylinder(1,d=20+c, center=true); 
translate([0,1.5,-3]) cylinder(1,d=20+c, center=true); 
translate([0,1.5,0]) cylinder(1,d=20+c, center=true); 
translate([0,1.5,3]) cylinder(1,d=20+c, center=true); 
translate([0,.5,6]) cylinder(1,d=20+c, center=true);

}
}

The problem is that it's no longer precisely the same shape. There are small
holes in your walls. You could make the cylinders smaller instead of bigger,
but you'd get places where the walls are very thin, which would be a problem
for some materials.

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Thanks for correcting the $fn to the right value as well. Now my cylinders
have nice continuous curve :)

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