Mean and SD for a normal distribution given value of a percentile Is it possible to calculate the mean and standard deviation of a normal distribution given a value x is the 25th percentile?
It feels like this is enough information but I can't seem to figure this out.

No, this is not possible: as whuber notes, you are trying to determine two parameters (the mean and the standard deviation), having only one data point.
For instance, a 25th percentile at $1$ can come from an uncountably infinite one-parameter family of normal distributions, because for any mean $\mu>1$ (do you see why $\mu$ must be larger than $1$?), you can find a standard deviation $\sigma$ such that $q_{0.25}(\mu,\sigma^2)=1$. A few examples, with the 25th percentile at $1$ marked in red:

R code:
perc <- 1
means <- c(2,4,6,8)
sds <- sapply(means,function(mm)
  optimize(function(ss)(qnorm(0.25,mm,ss)-perc)^2,interval=c(0.01,20)))

xx <- seq(-12,12,by=.01)+perc
opar <- par(mai=c(.5,.8,.8,0),mfrow=c(2,2),las=1)
    for ( ii in seq_along(means) ) {
        plot(xx,dnorm(xx,means[ii],sds[1,ii]$minimum),type="l",xlab="",ylab="",
   main=paste0("M = ",means[ii],", SD = ",round(sds[1,ii]$minimum,2)))
        abline(v=perc,col="red")
    }
par(opar)

Of course, if you have two (different) percentiles, you can determine a normal distribution (as long as the percentiles "make sense", i.e., they are not "the wrong way around").
