Will two distributions with identical 5-number summaries always have the same shape? I know that if I can have two distributions with the same mean and variance be different shapes, because I can have a N(x,s) and a U(x,s)
But what about if their min, Q1, median, Q3, and max are identical?
Can the distributions look different then, or will they be required to take the same shape?
My only logic behind this is if they have the exact same 5-number summary they must take on the exact same distribution shape.
 A: No, definitely not the case. As a simple counter example, compare the continuous uniform distribution on $[0, 3]$ with the discrete uniform distribution on $\{0, 1, 2, 3\}$.
A related example is the well-known Anscombe's quartet, where there are 4 datasets with 6 identical sample properties (though different from the ones you mention) look completely different. See:
http://en.wikipedia.org/wiki/Anscombe%27s_quartet
A: Just because the five-number summary is identical doesn't mean that the distribution is identical. This tells you just how much information is lost when we present data graphically in a box plot!
Perhaps the easiest way to see the problem is that the five number summary tells you nothing about the distribution of the values between the minimum and lower quartile, or between the lower quartile and the median, and so on. You know that the frequency between minimum and lower quartile must match the frequency between lower quartile and median (with the obvious exceptions, e.g. if we have data lying on a quartile, or worse, if two quartiles are tied) but don't know to which values of the variable those frequencies are allocated. We can have a situation like this:

These two distributions have the same five-number summary, so their box plots are identical, but I have chosen $X$ to have a uniform distribution between each quartile whereas $Y$ has a distribution with low frequencies close to the quartiles and high frequencies in the middle of two quartiles. Effectively the distribution of $Y$ has been formed by taking the distribution of $X$ and moving most of the data that is close to a quartile further away from it; my R code actually performs this in reverse, starting with the irregular distribution of $Y$ and levelling out the frequencies by reallocating data from the peaks to fill in the troughs.
EDIT: As @Glen_b says, this becomes even more obvious when you look at the cumulative distributions. I've added gridlines to show the location of the quartiles, which are the same for the two distributions so their empirical CDFs intersect.

R code

yfreq <- 2*rep(c(1:10, 10:1), times=4)
xfreq <- rep(mean(yfreq), times=length(yfreq))

x <- rep(1:length(xfreq), times=xfreq)
y <- rep(1:length(yfreq), times=yfreq)

ecdfX <- ecdf(x)
ecdfY <- ecdf(y)
plot(ecdfX, verticals=TRUE, do.points=FALSE, col="blue", lwd=2, yaxt="n", 
    main="Empirical CDFs", xlab="", ylab="Relative cumulative frequency")
plot(ecdfY, verticals=TRUE, do.points=FALSE, add=TRUE, col="black",
    yaxt="n", lwd=2)
axis(side=2, at=seq(0, 1, by=0.1), las=2)
abline(h=c(0.25,0.5,0.75,1), col="lightgrey", lty="dashed")
abline(v=summary(x), col="lightgrey", lty="dashed")
legend("right", c("x", "y"), col = c("blue", "black"),
       lty = "solid", lwd=2, bty="n")

par(mfrow=c(2,2))
hist(x, col="steelblue", breaks=((0:81)-0.5), ylim=c(0,25))
hist(y, col="grey", breaks=((0:81)-0.5), ylim=c(0,25))
boxplot(x, col="steelblue", main="Boxplot of x")
boxplot(y, col="grey", main="Boxplot of y")

summary(x)
#   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#   1.00   20.75   40.50   40.50   60.25   80.00 

summary(y)
#   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#   1.00   20.75   40.50   40.50   60.25   80.00 

A: This is most clearly answered by considering the (cumulative) distribution function.
Specifying the minimum, maximum and the three quartiles specifies exactly 5 points on the cdf, but the cdf between those points may be any monotonic nondecreasing function in between that still passes through those points:

In the drawing, both the red and black CDFs share the same minimum, maximum, and quartiles, but are clearly different distributions. Clearly any number of other CDFs could be specified that also pass through the same five points.
In fact, all we've done is restrict our distribution function to lie within four boxes:
$\qquad$
(as long as it also continues to satisfy the other conditions for a CDF). That isn't all that much of a restriction.
The same notion can be applied to sample quantities - two different empirical CDFs may nevertheless have the same five-number summary.
On that subject, see the four examples near the end of this answer, which all have the same five-number summaries, but which have very different looking histograms (which I'll reproduce below):

This again emphasizes that five number summaries don't generally do very much to tell us about shape.
