Zion Oil & Gas Inc (ZNOG)

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if an entity otherwise is not required to report stock ownership to the sec ,once they pass the 5% threshold they have to report either a simplified schedule 13g or a full schedule form 13g

so yes they can run under the radar for various reasons until they own 5%
warren buffet does this himself- buy up to but under 5% until he is ready to be identified as an owner of such a companies securities
,in his case secrecy until 5% can be because his 5% reporting will trigger some other fund managers to buy and thus possibly significantly increase pps,which would then make it harder for warren to buy more -once the cat is out of the bag

also, if somebody wants a seat on the board or eventually plans on a partnership bid or other types of events,he may not want his interest to be immediately known while he strategizes or acquires other partners or bidders etc--i have no knowledge these things are transpiring re ZN - but these are examples of things which can occur


https://www.investmentfundlawblog.com/resources/investments-by-funds/acquiring-5-publicly-traded-company/

https://www.lexology.com/library/detail.aspx?g=41a16a1b-a159-409c-98ad-39051b7f4753