There's quite a bit of missing context and misconceptions about this in general, but there is a theoretical backing that is really just a mathematical fact that isn't falsifiable by observation.

Bias, variance, and mean squared error for an estimator can be related quite simply. Say the quantity to be estimated is y, the estimate is x (forgive the lack of LaTeX making more conventional names difficult). Then MSE is E[(y - x)^2] = E[x^2] + E[y^2] - 2 E[xy] = Var[x] + Var[y] - 2Cov[x,y] + (E[x]-E[y])^2, and the first 3 terms are typically simplified by assuming some error structure that makes Cov[x,y] = 0, leading to a decomposition of MSE into 3 terms:

1. Var[y] - the variance of the true underlying quantity that is inherent to any estimator.

2. Var[x] - the variance of the estimator regardless of bias. (What's typically referred to as the variance in the bias-variance terminology)

3. (E[y] - E[x])^2 = Bias[x]^2 - the squared bias of the estimator.

Since (1) is constant across all estimators, that means that the difference in MSE for two estimators comes entirely down to (2) and (3).

The bias-variance tradeoff is then:

For a fixed value of mean squared error, decreasing the bias of an estimator must increase the variance by the same amount and vice versa.

An unbiased estimator will have the maximum variance among all estimators with the same MSE. It's entirely possible to improve an estimator's MSE by increasing or decreasing bias. In fact, the two commonly used estimators of sample variance are a great example*.

The most important pieces here are that we're talking about MSE of an estimator, and that for a fixed value of MSE there's a 1:1 tradeoff between bias and variance. It makes no statements about the case when MSE is not held constant, and it's actually very well known in mathematical statistics that biased estimators can provide significant improvements in MSE over unbiased ones. This doesn't hold if you're not considering MSE, or allowing MSE to change. It can be extended to multivariate cases and "estimator" could be anything from a mean of a few samples to the output of a deep belief network.

* Take a = 1/(n-1) * sum(x^2) - 1/(n*(n-1)) * sum(x)^2, and b = (n-1)/n * a. v = Var[x] is the population variance. We have E[a] = v, which means a is unbiased (this is Bessel's correction). And obviously, E[b] = (n-1)/n * v. Thus Bias[b] = E[b] - v = - v / n. Calculation of Var[a] requires knowledge of the 4th moments of x, however, in the case of normally distributed data the MSE of b is less than that of a. And a property that holds in general is that Var[b] = ((n-1)/n)^2 * Var[a], and MSE[b] = Var[b] + (v/n)^2 = 1/n^2*((n-1)^2 Var[a] + v^2). Thus MSE[b] < MSE[a] if (2n - 1)*Var[a] > v^2. This is true for normal distributions and many others.