Accident benefit with no delay
The accident part can start when the policy takes effect while illness coverage still has a waiting interval.
A specific waiver
An exam or qualifying offer may remove a particular wait. The waiver’s scope and activation requirements matter.
Routine-care benefits
A wellness allowance may be usable from the effective date. That says nothing about treatment for a new illness.
A continuing policy
Keeping an existing policy active is different from replacing it. Ask whether new benefits or a change of insurer bring fresh conditions.
Treat the phrase as an incomplete sentence: “No waiting period for ___, beginning at ___, under ___.” If the seller cannot fill those blanks with the offered documents, you still do not know what you are buying.