Yes, the board can raise your assessment
It is uncommon, it is real, and it is the reason to check whether you are the low one first.
An Illinois Board of Review has authority to increase an assessment as well as reduce it. Filing opens your assessment to review — not only the part you complained about.
How often it happens
Rarely. Most appeals result in a reduction or no change. But "rare" is not "never", and the cases where it happens are predictable: the review turns up something the record did not have. An addition never permitted. A finished basement. A square footage lower on file than in reality.
Who should be careful
Anyone whose property is better than the county's record of it. If you finished the attic without a permit, an appeal invites someone to look. Fix your expectations before you file.
The check worth doing first
Compare yourself to properties the county records describe the same way. If most of them are assessed more than you, you are already the beneficiary of the spread and filing is a bad trade. Any honest analysis will tell you this, and a tool that never says it is not helping you.
How to reduce the risk
Make sure the county's record of your property is accurate and that you would be content for it to be verified. Argue a narrow, specific ground rather than "my assessment is too high generally". And do not file for a saving so small it is not worth the exposure.