Dan and Michael point out the relevant issues. Just for completeness, the relationship between adjusted $R^2$ and $R^2$ is given by (see, e.g., here)
$$ R^2_{adjusted}=1-(1-R^2)\frac{n-1}{n-K}, $$ (with $K$ the number of regressors, including the constant). This shows that $R^2_{adjusted}=1$ if $R^2=1$, unless (see below) $K=n$.
$R^2=1$ occurs when all residuals $\hat u_i=y_i-\hat y_i$ are zero, as $$ R^2=1-\frac{\hat{u}'\hat{u}/n}{\tilde{y}'\tilde{y}/n}. $$ Here, $\hat u$ denotes the vector of residuals and $\tilde y$ the vector of demeaned observations on the dependent variable.
Dan discusses one reason to get an $R^2$ of 1. Another is to have as many regressors as observations, i.e., $K=n$.
Technically, this is because the $n\times K$ regressor matrix $X$ then is square. The OLS estimator $\hat\beta=(X'X)^{-1}X'y$ can then be written as (assuming no exact multicollinearity) $$ \hat\beta=(X'X)^{-1}X'y=X^{-1}{X'}^{-1}X'y=X^{-1}y $$ so that the fitted values $\hat y=X\hat\beta$ are just $\hat y=XX^{-1}y=y$, so that all residuals are zero.
Here is an illustration using artificial data (code below), in which regressors are generated totally independently of $y$, and yet we achieve an $R^2$ of 1 once we have as many of them as we have observations.
Code:
n <- 15
regressors <- n-1 # enough, as we'll also fit a constant
y <- rnorm(n)
X <- matrix(rnorm(regressors*n),ncol=regressors)
collectionR2s <- rep(NA,regressors)
for (i in 1:regressors){
collectionR2s[i] <- summary(lm(y~X[,1:i]))$r.squared
}
plot(1:regressors,collectionR2s,col="purple",pch=19,type="b",lwd=2)
abline(h=1, lty=2)
When $K=n$, R however, correctly, does not report an adjusted $R^2$:
> summary(lm(y~X))
Call:
lm(formula = y ~ X)
Residuals:
ALL 15 residuals are 0: no residual degrees of freedom!
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.36296 NA NA NA
X1 -1.09003 NA NA NA
X2 0.39177 NA NA NA
X3 0.19273 NA NA NA
X4 0.51528 NA NA NA
X5 -0.04530 NA NA NA
X6 -1.28539 NA NA NA
X7 -0.72770 NA NA NA
X8 -0.14604 NA NA NA
X9 0.34385 NA NA NA
X10 -0.93811 NA NA NA
X11 2.23064 NA NA NA
X12 0.06744 NA NA NA
X13 0.21220 NA NA NA
X14 -2.29134 NA NA NA
Residual standard error: NaN on 0 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: NaN
F-statistic: NaN on 14 and 0 DF, p-value: NA