Are there any circumstances under which an accurate measurement may not be precise?

Depends on what you mean by precise.

Precision can either be the number of decimal places to which you are sure, or the consistency of multiple trials.

PRECISE (CONSISTENT) BUT NOT ACCURATE?

For example, you can be extremely precise but extremely inaccurate, if you have a consistent error that you make, such as not subtracting the mass of a watch glass and filter paper when you really want the mass of the precipitate on filter paper on a watch glass.

Some data of that sort might be:

#"48.364 g", "48.372 g", "48.351 g"#

when the actual mass of the precipitate might be, say, #"1.052 g"#, #"1.060 g"#, and #"1.041 g"# for trials 1, 2, and 3 respectively, as the typical watch glass is around #"45 g"#, and the filter paper is the rest.

ACCURATE BUT NOT PRECISE (CONSISTENT)?

It is not possible to have many accurate measurements that are also not precise.

When you are accurate, you must be getting the correct answer on each trial. Otherwise, you are not accurate, and when you are not accurate on each trial, you are thus not precise (consistent) either.

For example, if you get #"1.052 g"#, #"1.053 g"#, and #"1.051 g"# for your precipitate, that's quite precise. It's also quite accurate, if that's what you're supposed to get.

But if one of those data points was #"2.000 g"#, then you are no longer consistent by virtue of being inaccurate in one trial.

ACCURATE BUT NOT PRECISE (CERTAINTY)?

But, if you mean precision in terms of decimal places to which you are certain, then yes, it is possible. Just be bad at writing out the proper number of decimal places.

When a scale gives you 3 decimal places for your mass, write all 3 down. If you don't, you are ignoring the precision (un/certainty) you could have attained.