I recently completed a mission for Yes Man and got both the above mentioned mission and a similar one from Caesar's Legion

I don't want to damage my reputation with the NCR, so how can I prevent this from occurring?

You can only progress so far with any one faction before the others turn against you.

Thats why you have to choose.

As Deven said, it's a warning quest, you can do up to 2-3 main quests for one faction before they turn hostile. Yes-Man is more or less the 'back up' faction.

Work for NCR? Mr. House and Legion wont want you around. Piss off NCR down the road? Yes Man is your only option left.

Hostile isn't the right word. Depending on your standing with the factions, you could still be on good terms with them, they just don't trust you to work for them.

Anyway, I never had a reputation really go down the drain by failing the quests, though the NCR loved me and the Legion hated me which might have had an influence.

The Legion will actually turn irrevocably hostile to you if you work too far with the other factions, regardless of your standing with them.

It's fairly straight forward, apart from the Legion, most of the quests leading to the end are interchangeable so you can choose who you do them for and keep in everyone's good books until you have to make a real choice. For instance, House wants you to wake up the robot army and Caesar wants you to use the chip and blow up whatevers in the vault and you can do both. Caesar will then ask you to do something else and let you go but you don't actually have to do anything, just leave the fort. Yes Man wants you to investigate various factions and so does the NCR, so do both at the same time. The Legion will fall out with fairly quickly and you'll have to support or kill House but you can right to the end before the NCR gets the hump.

Perhaps that wasn't quite as straight forward as I thought.

bwilderbeast CL is possible at the same time as Yesbot, and some NCR.
If you cherry pick quests and make sure you juggle to CL on the strip, and ignore the more anti legion ones off it..

Again though if not actively working for CL they'll be the first to turn on you.
Late game last mission with thought but still the instant hostillity.