Well, it can happen if the clearing corporation fails to manage risks, but it hasn't happened in the history of our capital markets (To the best of my knowledge). However, your concerns are legitimate, and I'll try to shed light on this.
A clearing corporation is a Central Counter Party (CCP), and as a result, it is exposed to the maximum systemic risk among all stakeholders in the trading ecosystem. Why? Because they are liable to settle the trades if there are defaults from clients, brokers, or clearing members. Since the clearing corporation (CC) guarantees settlements, they are required to maintain a sufficiently funded "Settlement Guarantee Fund" and "Default Waterfall" to pay the settlement obligation in the event of defaults.
If the default amount is greater than the reserve funds, the CC can go bust. However, it is unlikely because there are layers of regulations and laws that protect the CC. For instance, the TM will be held liable if a client defaults. If a TM defaults, the CM will be held responsible, and if the CM defaults, the liability will be shifted to the CC. Even in such scenarios, the CC calls the shots greatly and can annul the trades to prevent a settlement (If there is any merit in doing so). This was about to happen in 2019 when a UP-based broker apparently pledged Dalmia Bharat MD's MFs worth 400+ Crores to take trades in F&O without prior consent. The fiasco led to chaos in the broking fraternity as the largest clearing member, IL&FS, got involved, and subsequent events led to its untimely shutdown. From what I hear, NCL was relatively unaffected by this.
Based on the risk-based requirements, the size of the settlement guarantee fund is enhanced. I vaguely remember SEBI asking NSE to hike the amount by 3-4x to be able to handle future settlement risks. Also, a Clearing Corporation has access to a funding window from the Reserve Bank of India (RBI) to the best of my knowledge. So, it's unlikely that they will default. After my interactions with the senior leadership at clearing corporations, I am confident that they can handle crises effectively, as they have done in the past.
I hope this answers your question.