A crime commited while under the influence of drugs and/or alcohol. High crimes are usually fairly harmless and poorly thought out. They are met with regret or a completely loss of memory of their occurance the next day. Victims of these crimes are usually perplexed as to why they were targeted. Vandialism, theft, and verbal/physical assult are most common.
Guy: "This random guy started talking shit to me at the bar then shoved me!"
Friend: "Sounds like you were a victim of a high crime."
Homeowner: "I went outside this morning and my lawn furniture was missing!"
Officer: "I've seen alot of these, it was probably a high crime. Nothing I can do."