Is bias a property of the estimator, or of particular estimates? As an example, I often encounter students who know that Observed $R^2$ is a biased estimator of Population $R^2$. Then, when writing up their reports, they say things like:
"I calculated Observed $R^2$ and Adjusted $R^2$, and they were pretty similar, suggesting only a small amount of bias in the Observed $R^2$ value we obtained."
I get that generally when we talk about bias we're typically talking about the properties of estimators rather than particular estimates. However, is the quoted statement above a misuse of terminology, or is it OK?
 A: In statistics, bias is clearly a property of the estimator.
I share your observation that bias is often incorrectly applied to estimates. Your example seems rather innocent in that regard, because a well-meaning instructor could argue that your students assumed that the error of the estimates is so small that it's OK to equate the estimate with the estimator. 
A more extreme example would be the use of the word "bias" for the error of a particular estimate, as in: we know the true value is 5, but our estimate was biased upwards. I feel this is indeed a misuse of terminology that will eventually lead to confusion, and one should therefore flag it as inappropriate. 
A: Bias is the property of an estimator. 
An estimator is itself a random variable and has a distribution (with a mean and variance). When an estimator has an expected value that is equal to the true, unknown value that it's trying to estimate we say the estimator is unbiased. 
Now, when we calculate an estimate we are looking at one observation from the distribution of the estimator. So, even if we go with the (incorrect yet innocuous in this context) definition of bias the student seems to be using, there's a problem. The single observation (the estimate), may be very far from the expected value of the estimator's distribution. In other words, it's possible the value of the estimate is very far from the true underlying value where the student seems to be implying that the Observed $R^2$ is very close to it's true value.
