Given only the Mean, Q1, Q2, and Q3, is there a way to predict the value that would fall at the 60th percentile? It's been a while since I've had to mess around with statistics like this. I only have the mean, 25th percentile, median, and 75th percentile. I am attempting to predict with reasonable accuracy what value would fall at the 60th percentile. Is this possible with the data I have available? Or would I need more?
An example of the data would be something like this:
Mean - 456,000 ;
25th Percentile - 328,000 ;
Median - 384,000 ;
75th Percentile - 516,000.

Without more information/assumptions, the quartiles alone only restrict the 60th percentile to be between (or equal to) the quartiles either side of it (in this case the 50th and 75th percentiles).
The mean potentially adds information but typically won't provide enough additional information to constrain the value further (sometimes it might, particularly if it can be combined with still other information, but typically there will be possible arrangements of data values that still allows the 60th percentile to reach both quartile limits).
In the case of your specific example, the mean doesn't appear constrain the possible value of the 60th percentile beyond it being in the interval $[384000, 516000]$. Without more information (perhaps including the things I specifically asked about in comments), it looks like both those endpoints are possible.
For example, here's a case (of invented data) where the 60th percentile is the same as the 75th percentile (values are in thousands):
summary(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  328.0   328.0   384.0   465.0   516.0   671.8 
quantile(x,.6)
60% 
516 

