Crysis

Just adding something to the above. I did the Select Level thing and like Brum said it was a no go for these achievements. I did the New Game thing which did work for Brum but not for me. If you have the same problem I found a way around the glitch. You need to go into your Xbox Memory file (It is a few windows into your Settings Window, Last Window in your Gamer Row) and delete your saved game if it was done at a lower difficulty. You will keep all the weapons attachments you collected if you do this. Now reload Crysis and before you start a new game make sure you go into options and select Delta then go to Campaign and select new game and again select Delta. This should work if you are having problems getting the achievements to pop. You can check if it worked by quitting the game and then selecting the select level choice and seeing if the Delta Symbol is in the Upper left hand corner of the level pic, if so the game is finally recording Delta wins (if you have beat the level/game on Delta but no achievements when you check this you will see the symbol is for what ever difficulty you first beat the level on and did not change when you beat it on Delta, this is the Glitch). So clear memory and start again.
Couple of other points/tips.
1) You can skip Cutscences, will not affect achievement. Saw posts saying it did, I skipped and got all 4 achievements.
2) You DO NOT need to do the secondary missions to complete the levels, just primary. Use vehicles and race to the checkpoints, you Do NOT need to engage all enemy positions, just get you ass to the end of the level.
3) AI is stupid, if you can get yourself into a building or trailer using cloak in camps do it, the guys just funnel into the doorway, except maybe 1 or 2, easy kills.
4) Best Tactics, hide, CLOAK, and come around your cover slightly, pick you target (head shot, dead right now) then get back behind cover, wait for energy to refill and rinse and repeat). CLOAK will save you ass, use it. Guns should be 1. Enemy AK47 (forgot the real name) 2. Sniper rifle when you can get it, till then then your first assault rifle,. The shotgun not good a distance and once you are in close you are DEAD. Was not a fan of the sub-machine gun. CONSTANTLY pick up ammo, I do not know how often I found myself LOW and in a huge battle till I started doing this, even pick up ammo to the max for weapons you do not have as you may need it if you run out in one of your guns and can find one of these weapons lying nearby. Get ambushed, hold down B and lie flat on the ground, smaller target and you might just survive.
5) Hardest part for me was the last boss. Best approach is take out his 4 guns up close using the large wreckage right in front of him and use mini gun on turrets (so pick one up as well as a Guass Rifle and ammo, this will be all you need). Once they are down run back to the door (now blocked) that you came out of to get onto the deck and there is more mini-gun ammo there, pick it up and now run to you left around the wrecked fighter and through the fire (you will take damage but not die) and keep moving to your left and back and you will see 2 more wrecked fighters near the edge run to these and over the edge you will see a walkway that you can jump down to, get down there, crouch down and wait. Psycho and the girl in the VTOL will talk to you, wait till they ask for help then CLOAK and stand up, the aliens will not see you, line up the closest one and let him have full Mini-gun carnage and when he goes down quickly crouch down again and max out you health and energy then rinse and repeat on the next two aliens. If the girl says use the Tac Gun do not do this till all 3 aliens and down and dead do not worry the shield will stay down till you are ready. Also do not worry about the VTOL, the aliens cannot destroy it. Once one side is down it will repeat the same 3 alien thing again so do the above again. Stay on this walkway till all the aliens are dead then jump up and put the second Tac Gun round into the other side. Checkpoint here. Now Equip the Gauss rifle and run up to the Alien ship, it will begin to rise up and fly over the ship deck. On the other side of the ship is another walkway lower down, get down there and depending on which side of the alien ship you are use the Gauss to take down the turrets first on your side then start blasting the circular lit up hatch on the underside with the Gauss as well (should take several blasts to do this and open a hole). if small aliens attack you just use the Gauss on them as well. Once the Hatch is open put a TAC round inside and you have taken down the alien ship. Now jump up and race to the end of the deck for pick-up, hide behind the raised jet blast panel near the back of the deck and equip the mini-gun again will full ammo. Hiding here will delay the small aliens from tracking you, use cloak if you have the energy left. Just stay here and kill any aliens that do find you and wait till the VTOL has gotten into position at back of ship then race to the back and get on board. Congrats and enjoy the Cutscene (double achievement time for 120 points).
5) Just to say it, take out enemy at long range if you can.

Good Luck. not really super tough you just have to avoid encounters when you can and either drive into bases then quickly find cover or go around them, lots of bases/roadblocks can be skipped or avoided al together.

Be warned. I just did these levels on Delta via the Mission Select Menu after completing the game for the 1st time on easy and it did not count. My advice would be to start a new game and set it to delta otherwise you may suffer the same problem as me. Check the mission select screen after each level to make sure it is being completed on Delta.

*EDIT* I just completed "Contact" on Delta again after Starting a New Game and it has counted. Again i would advise against using the Mission Select.

Please see "Glenn 46's" solution below for more information regarding the save game glitch that may occur. We only want to save you the effort of having to play levels repeatedly, hope this helps!