Companies are considered "non DTC-eligible" if they are not eligible for DTC FAST (Fast Automated Stock Transfer) which allows electronic settlement in book entry form. Non FAST-eligible stocks must be settled manually with certificates, which is more costly for the clearing firm, transfer agent and brokerage.
The DTCC does not issue legal notices for companies that never had or later lose FAST eligibility as they do with companies that receive DTC Global Locks (complete suspension of all services except custody), a chill on a certain service such as custody or a company for which the NSCC exits its position from the CNS (Continuous Net Settlement System). The last group are designated trade for trade (T4T) which means that the DTCC will not be involved in settling trades. T4T trades must be settled directly between the buying and selling brokers.
Although companies with global locks and those designated T4T are obviously non DTC-eligible, so are the many companies that have never been FAST-eligible or lose FAST eligibility with no legal or public notice. That's why brokers like Zecco list many more stocks as non DTC-eligible than those for which legal notices have been issued.
It's important for investors and traders to be aware of the non DTC-eligible companies that have never been mentioned in a legal notice as some brokers restrict trading or charge additional fees to trade them due the extra costs and fees involved.
The bolded part of the quotation indicates things that can hamper intitial or threaten continued DTC eligibility.