I've just been learning to play the acoustic guitar for 5 months. Now I want to play the blues in the traditional delta Blues style. Is it possible for me to play one or two simple blues songs fluently in just 4 months?

My guess is your teacher isn't particularly experienced himself with delta blues and has a personal insecurity about teaching the style. However, he also has a better understanding of your current skill than anyone commenting here and, quite possibly, even you. I'd trust him on this one. Certainly, you should be able to focus your studies and practice time on any genre that interests you but there are some basics to master before you go trying anything more advanced.

if your retention is good, you should be able to get a strong understanding of a basic blues style in a week. For instance, bottleneck slide in open D tuning is essentially very simple: three chord shapes, all easy to use barred, plus slide with vibrato, all in I, IV and V position. The bottleneck isn't essential, but it sure is fun. If you mean "able to play a song by John Lee Hooker note for note" you may never get there... but four months should be enough time to be very, very satisfied with what you can play. Here's a really sweet ten minute lesson on YouTube: "Slow Blues in E" taught by Stefan Grossman: I've been playing blues off and on for over forty years, and there's licks in there I intend to use... If you can learn these, the Blues is your oyster.

If you like the Delta Blues sound you might want to explore open turnings and slide.  Here's a good beginner lesson. There are 1000's of good examples online.  Have fun with it!

Of course it is possible - if you are relaxed about the meaning of the word "fluency". Your teacher suffers from a lack of imagination, or perhaps they are inflexible in their standards. Just do your best. Learn as much as you can, and strive for ease and relaxation after you learn the basic grips. Then after 4 months give your best performance and drop the mic.

A few people have said it and I'll restate it: don't listen to your teacher when he says it's impossible. Perhaps he believes it is impossible because he himself does not play the blues? Get a book with a bunch of blues songs tabbed out. DO NOT get a "beginner" book on blues--they are easier for sure but you can teach yourself all that stuff pretty quick if you're diligent. You want something that's fun to play and you're not going to find it in a beginner book. Find all the songs on YouTube and listen to each of them. You'll almost certainly find many versions of the song ; listen to a good handful of the versions and listen to each of these a few times. You'll listen to each song in that song book a good 15-20 times. This will be maybe an hour spent just to listening to each song, likely 10-15 hours listening to the whole book. You really want to understand the songs, predict the movements, know and feel the lyrics. From all of these songs, you'll surely have two songs you really dig so dive in there. Don't expect to be able to play straight through the song. Take it bar by bar. Any time you struggle with a chord change, practice that change. Any time you struggle with a lick, practice that lick. Any time you struggle with anything, practice that thing. If you have a question on how to do something, look it up on YouTube or ask a question here or some other guitar forum. Practice starting from every bar and playing a few bars. Maybe play it backwards. You want your brain to have wired into muscle memory all of the physical movements that are required to make all the various sounds of the song. And this takes time. Prepare to commit to this practice regime for an hour a day, one month per song and you'll really impress yourself. An hour a day may sound hard. But if you dig the song and you stick with it a bit, you'll undoubtedly progress. And once you start to see yourself progress you may well curiously find that your days offer more and more time to play. Welcome to loving music. If that happens, be mindful to take breaks :) No point goin and injurin yourself now