Printing presses, no matter how advanced or how expensive, have slight variations during the printing process. This means that each sheet of paper or card can move ever so slightly as it's moving through the machine. This slight shift is barely noticeable in normal circumstances, but with a thin border, it can make it leap off the page. A 2mm border, shifted by 1mm, would give 3mm on one side and 1mm on the other. This extreme example gives an idea of what can happen.
After printing, the paper goes through a cutting process. Just like printing, cutting also has tolerances, meaning the cut might differ slightly from where you expect it to be. Paper is cut in a large stack with an industrial guillotine so the paper can shift slightly as the blade moves through the pile. A border too close to the edge can be partially cut off, resulting in a lopsided appearance.
When you add up both of those tolerances, you can see why it's challenging to maintain an exact thin border on multiple sheets. Advanced ways to cut paper, like using a laser, are available, but they increase the cost and are usually reserved for super high-end products.
Always maintain a safe margin inside the edge of your document. A general rule of thumb is to keep all critical elements, including text, logos and borders, at least 6mm away from the edge of the document. This safe margin ensures that your design remains intact and visually appealing, even with slight shifts. To continue the example above, a shift of 1mm on a 6mm border gives 5mm on one side and 7mm on the other. Whilst it could be a better look, it wouldn't be as noticeable as a 2mm border.
Set up your document with the appropriate bleed. Typically, a 3mm bleed on all sides is recommended. This means your design should extend beyond the final cut size by this amount. For example, if you're designing an A6 postcard (105 x 148mm), your document size should be 111x154mm, allowing for 3mm bleed on each side. With some software like Adobe Illustrator or Indesign, you can set the page size and separate bleed sizes, but with Photoshop, you need to do the calculation yourself.
If you must include borders, make them thicker and position them well within the safe margin. Small borders are more likely to appear uneven due to printing and cutting tolerances. A thicker border has a better chance of looking even and, with that, more professional.
Different printing companies may have varying requirements and tolerances, so what might be good for one set of machinery might not be good for others. When in doubt, get in touch. It's much easier to get it right at the start then to go back and try fix after it's been signed off