A long time ago I made a thread which asked why do you think Tolkien never mentioned anything about what we call the "common cold" on M-e, in specific with the journey of The Fellowship. The answers we gathered were
1) M-e air was clean and "colds" were non-existent there
2) Tolkien doesn't write about them going to the bathroom, so he isn't going to bother writing about a character having the sniffles.
Either way you look at it, I could imagine that some sort of physical weakness that resembles our "common cold" had to have been taken towards Frodo a little. There's no evidence for this, and maybe he didn't have anything like our cold, but this passage in
The Stairs of Cirith-Ungol even gives me the shivers-
"The passage seemed to go on for miles, and always the chill air flowed over them, rising as they went on to a bitter wind."
The hobbits were already probably sweating bullets. I can't see how Frodo couldn't have, along with the burden and weakness he already felt from the Ring, had a little extra "cold". I pick Frodo because, in his already weakened state, it would be more feasible that he got the "cold" rather than Sam.
I'm sure both reasons 1 and 2 I mentioned are correct, and maybe I'm looking
a little too deep here. Just thought it was worth a mention.