First, let's keep things simple.
The most common keys in old timey, folk, or bluegrass jams are C, D, G, and A. With a standard-sized dulcimer tuned to an open D (DAA or DAd), you can obviously play in the key of D. You can capo at 3 to play in G or capo at 4 to play in A. And you can tune down to CGc or CGG to play in C. (And there are other ways to get those keys as well, without switching instruments. You could, for example, tune DGd to play in G, although if you chord and/or fret across all the strings that tuning will require some adjustments.)
So for 90 percent of your playing--even in multi-instrument jams--you only need one dulcimer.
However, I will be honest that I often bring 2 dulcimers to jams. One a standard dulcimer and one in the baritone range. The standard size dulcimer can usually be tuned anywhere between C and E. The baritone can usually be tuned anywhere between F and A. If we add the half steps in between those ranges, we get 10 keys, the only ones missing are B and Bb. And you can finagle (is that how you spell that word?) around to get those keys as well. For example. if you tune a standard dulcimer up a half step to Eb, you can capo at 4 to get Bb.
So there is no need to get multiple dulcimers for multiple tunings and multiple keys. Even without using a capo, you can get 10 of the 12 possible keys with dulcimers in two tonal ranges. (One of my baritones can get up to Bb as well, but be careful and wear protective goggles if you try that on your own.)
However, I have several dulcimers. Why? I find different kinds of playing sounds better on different kinds of dulcimers. I love my Blue Lion for fingerpicking, but don't think it sounds so hot flatpicking. I love my Modern Mountain Dulcimer for flatpicking, but find the action is too low and the strings too close together for fingerpicking. I have one dulcimer that originally came with six strings, and it sounds great on those tunes with tons of strumming. But playing more nuanced arrangements with lots of precise picking and lots of hammer-ons and pull-offs, those double strings just don't do the job.
So don't go buying different dulcimers for different tunings or different keys. It's completely unnecessary. However, you may find that as your playing evolves, you develop different preferences and one or another dulcimer may be great for one tune played in a certain style but not for other tunes. Getting multiple dulcimers to please our preferences, though, is a luxury, not a necessity.
Dusty T., Northern California
As a musician, you have to keep one foot back in the past and one foot forward into the future.
-- Dizzy Gillespie