Can my internet access be blocked by DNS servers not responding to my computer? If so what can I do to get around the problem?
Practically yes - DNS is used to convert a domain name into an IP address of which server to go to - so if DNS does not resolve then the computer can't reach out to the server to get the information it requires.
The solution depends on how the network is configured.
The simple answer is to set up alternative DNS servers - 184.108.40.206 (Googles) and 220.127.116.11 (Cloudflares) are 2 common, open and free DNS servers. Of-course, if your network is blocking DNS to servers other then the one you are receiving in DNS then this won't work.
A possible workaround would be to turn on "DNS over HTTPS" if thats available to you (you should google that). This bypasses the standard DNS network and makes DNS requests over HTTPS. (Have a look at https://www.howtogeek.com/765940/how-to-enable-dns-over-https-on-windows-11/ if you are using Windows)
If this all fails you likely want to look at some kind of VPN service.
(If the problem is only for a few sites, you may be able to get their IP addresses and hardcode them into your hosts file - but that is not a good or long term solution)