Sometimes. But astronomers usually work so furiously to publish their research that the chances of this happening are small. I know of several instances where some astronomers have taken too long to publish, and other astronomers have retaken their data and published their own conclusions. This happened to me a few years back. In 1990 I used the Kitt Peak 12-meter telescope to observe carbon monoxide molecules in a distant interstellar cloud. It took me several years to write up this research since I was involved in other activities. Just as I was getting ready to submit the article to the Astrophysical Journal, I noticed that another astronomer had just published essentially the same data on the cloud which he and his colleagues had taken with the same telescope!
For large projects like the Hubble Space Telescope, the chances are much smaller, however the Keck Telescope is a rival to the Hubble Space Telescope in several important areas. If I was an observer on the Hubble I would be constantly looking over my shoulder at the Keck, and in fact, to get an observing proposal accepted on the Hubble, you have to demonstrate that the same research absolutely positively cannot be undertaken on ground-based telescopes. This reduces the chance that any observation on the Hubble will be made obsolete by similar ones at the Keck. The only way that space astronomy data can become obsolete is by other space astronomy projects which take 5 - 10 years to plan and carry out as 'follow-ups' to the original observations. Even now, the Einstein X-ray observatory data is still in use even though it is 15 years old, and the infrared data from the Infrared Astronomical Satellite is over 10 years old.
So, the chances that data become obsolete before it can be properly investigated is probably less that 1 percent for space-based data, but can be much higher than this for ground-based optical and radio data, although the judges for observing proposals always make certain that the prospective observer is not usually reproducing exactly someone elses prior observations.