Can an Army Infantryman wear both the EIB and CIB on the uniform? What happens if you get both?

One final point you can get killed earning the CIB

SFC Severs the EIB is a challenge and the CIB is not? Ranger Inf Advisor Vietnamese Army ‘ 68 - ‘ 69 mine was not 100 hour war or ‘ Circumstantial’In 68 and 69 we lost over 27,000 guys. Your comments are ignorant outrageous and damn insulting

Can’t understand if you have been in combat why you would wear EIB? CIB is the real deal, what you train for

OK, the EIB says you are a well trained troop. Troops with less training have not yet earned it. I get that. It is fine for a troop to take pride in achieving any difficult accomplishment. I also understand there are a lot of fine troops, with varying degrees of training and experience, who have never been sent into combat. You can be a gung-ho super-troop, but if you are never in combat, you will never earn a CIB.
All that said, I have been through quite a few schools and was honor grad several times. I have several rows of medals - for going where I was sent, and doing what I was told.
I only earned three distinctions in combat and those are the ones I am proudest of. I wear my CIB, usually on my VFW Post Commander's cap since mine was in Vietnam and I have not been on active duty in many decades.
I only display ONE medal on my desk, my Air Medal, again because that's the only one that was for combat action. Finally, I am proud to have earned the right to wear a combat patch on my right shoulder. In fact, I have three of those to choose from, 101st Airborne, 2nd Infantry, and 18th MP Brigade. I am proud of all three and have fond memories of the men I served with earning the right.
So, if you earned both an EIB and a CIB, suck it up buttercup, the CIB is the big boy award, wear it with pride.

If you had earned and were awarded the CIB why would you want to wear the EIB. A close comparison of the EIB and CIB will reveal the CIB is essentially the EIB with a wreath around it. Now, I know what you are thinking, some people earning the CIB may actually not have earned to EIB, so be it but you might think if you were awarded both you should be allow to wear both. I guess in the same vein as the Airborne wings and the Glider or Ait Assault wings, in which case both are worn. As I think about this I would probably want to be able to display both. While similar they are indeed meant to represent two different accomplishments. So, I don't know. It must be an Army thing.
No disrespect intended.

You are proud of your EIB, but holding both you would wear the CIB. Never both.

Coming from the USCG I'm not familiar with the EIB. Can you describe the badge and how it's earned? What's the "E" stand for?

Why would one want to wear both?

No. You’d wear the CIB. However, even after earning your CIB, you’d still be strongly encouraged to earn your EIB.

If you were the EIB but you also have a combat patch, doesn't it follow that you have both? It's too badd that they are not both authorized to be worn simultaneously in my opinion.