Consider $n$ unit-variance random variables $X_1, X_2, \ldots X_n$ with the property that $\operatorname{cov}(X_i,X_j) = q$ for all $i \neq j$. Then,
the covariance matrix of these random variables is the same as the
correlation matrix. Now
$$\begin{align*}
\operatorname{var}(X_1+X_2+\cdots+X_n)
&= \sum_{i=1}^n \operatorname{var}(X_i)
+ 2\sum_{i=1}^n\sum_{j=i+1}^n\operatorname{cov}(X_i,X_j)\tag{1}\\
&= n + n(n-1)q\\
&\geq 0
\end{align*}$$
and so it must be that
$$q \geq -\frac{1}{n-1}$$
as Michael Hardy noted in a succinct comment on the question. The upper bound
is, of course, $q \leq 1$. Both bounds are achievable. Obviously, if
all the $X_i$ are the same random variable $X$, then $q = 1$.
For the lower bound, suppose that the $X_i$ are independent
unit-variance random variables so that they enjoy the desired constant
correlation with $q=0$. For each $i$, set $Y_i = X_i-\bar{X}$ where
$$\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i.$$
Then,
$$\operatorname{var}(Y_i) = \left(\frac{n-1}{n}\right)^2
+ (n-1)\left(\frac{1}{n}\right)^2 = \frac{n-1}{n}$$
while for $i \neq j$,
$$\begin{align}
\operatorname{cov}(Y_i,Y_j) &= \operatorname{cov}(X_i - \bar{X}, Y_j- \bar{X})\\
&= \operatorname{cov}(X_i,X_j) - \operatorname{cov}(X_i,\bar{X})
- \operatorname{cov}(X_j,\bar{X})+ \operatorname{var}(\bar{X})\\
&= 0 - \frac{1}{n} - \frac{1}{n} + \frac{1}{n}\\
&= -\frac{1}{n}
\end{align}$$
showing that all the correlation coefficients do indeed have
the minimum value
$$ \frac{-1/n}{\sqrt{(n-1)/n}\sqrt{(n-1)/n}} = -\frac{1}{n-1}.$$
Returning to $(1)$, note that if the correlation coefficients are
not required to all have the same value, then from $(1)$, we get
that the sum of the $n(n-1)$ correlations must be at least $-n$.
Thus, the average of the $n(n-1)$ correlations is at least
$-1/(n-1)$ and since at least one correlation must be as large
as the average, we can assert that
In any collection of $n$ random variables $X_1, X_2, \ldots, X_n$
with finite variance, there must be at least
one pair of random variables $(X_i,X_j)$ (with $i\neq j$) for
which
$$\operatorname{cov}(X_i,X_j) \geq -\frac{1}{n-1}$$