In abstract theory, yes, because of the correlation structure.
If you had uncorrelated measurements (with positive variance), then the weights could only be positive.
Example:
Let $\mu$ denote the true value. Let noisy measurement $X_1 \sim \mathcal{N}(\mu, 1)$. Let $X_2 = 2X_1 - \mu$, hence $\operatorname{E}[X_2] = \mu$ but $X_2$ is perfectly correlated with $X_1$. The covariance matrix is $ C = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} $.
The solution is $\mathbf{w} = \begin{bmatrix} 2 \\ -1 \end{bmatrix} $. Observe $2X_1 - X_2 = \mu$ and that linear combination has no variance (because $\mathbf{w}$ lies in the null space of $C$).
An equivalent finance problem:
An almost equivalent problem is solving for the minimum variance portfolio in finance.
Let $R$ denote a vector of $n$ returns. Let $C = \operatorname{Cov}(R)$ denote the covariance matrix of $R$. Let $\mathbf{w}$ denote a vector of portfolio weights.
The minimum variance portfolio is found by solving:
\begin{equation} \begin{array}{*2{>{\displaystyle}r}} \mbox{minimize (over $w_i$)} & \mathbf{w}'\Sigma \mathbf{w} \\ \mbox{subject to} & \mathbf{w}'\mathbf{1} = 1 \end{array} \end{equation}
This is exactly the same problem and it has exactly the same solution. For invertible $C$:
$$ \mathbf{w}_{mvp} = \frac{C^{-1} \mathbf{1}}{\mathbf{1}'C^{-1}\mathbf{1}}$$
Perhaps in the portfolio context, it's more intuitive that the minimum variance portfolio may involve both going long and short assets? (Note: before you run off and try to start an investment fund realize that estimating $C$ has big time problems.)
Some linear algebra interpretation
Let $U'\Lambda U = C$ be the eigenvalue decomposition of $C$. (This is basically PCA). Then $Y = R U$ is a random vector of uncorrelated random variables whose variance is given by the diagonal matrix of eigenvalues $\Lambda$. The minimum variance portfolio will give you positive weights on these random variables $Y_1, \ldots, Y_n$ but since these components are themselves linear combinations of security returns, you may get positive and negative weights in security weight space.