I would say there are not enough resources available on that modem/router and the dns service dies. But need to log in and check what's happening.

Are all the computers wireless? If so how many?

If any computers are wired do they have the same problem? Have you tried connecting to the router using an ethernet cable?

Yes, all computers are wireless. I did not try to connect via cable. When next time this happens, I will connect with cable and post result here.

One more thing, when the PCs are unable to browse (due to failing in making new connection), I can still browse with my mobile (which is also wireless). It means that a mobile can make new connection in this situation.

The strangest thing here is that if all laptops fail to browse then how mobile is able to browse the internet?

you should have told us that windows computers are also affected; obviously it can't be a linux thing then.

have we ruled out your ISP, and its DNS especially?
maybe it doesn't like "too many connections", and simply kicks you out.
maybe for some misguided reason it thinks the phone should not be kicked out.
or maybe the phone is indeed using google's DNS servers.
maybe you should try that on the other machines (why haven't you already)?

In my earlier posts I have mentioned that all laptops (linux and windows) are affected by this. You may have missed that...
I tried Google DNS in my router/modem setting but it did not help.
I am also thinking about resetting my modem to factory setting. However in this case I have call ISP and wait for long minutes in order to set my modem again. Thats why I am avoiding it...
I think the issue is with DNS. I will try Google DNS again but kindly guide me that would it make any privacy/security related issues?

I am not convinced it is a DNS issue. The wireless part of the router may not be able to handle the number of devices. The wireless part does not have unlimited bandwidth and connections. Knowing if it happens with a wired device would provide additional information.

I'm guessing that the phone is switching over to the cell network when the wireless isn't working for it. Recent iPhones are configured to do that silently out of the box, for example. I don't know if the icon even changes.

I still think there's a problem where the modem is losing the DNS information. That could be a hardware problem with the modem or it could be a problem on your ISPs end. Is the modem yours or leased from them? How old is it?

Edit: The overloaded wireless thought is possible, too. How many different devices are connected when this happens?

For this, we have to wait for next round of problem. Next time when it will happen, I will connect to wire and provide you more info...

No, phone is not switching to cell data becuase I have limited mobile data and I remain very careful about using it. Keeping it always off and only turning it when necessary. Moreover, I have android mobile, not iphone.
Yes, modem is from ISP and it is approximately 2-3 years old.
Approximately 8-10 devices simultaneously connect to wifi.

That seems a lot...looking forward to what happens when you hard-wire.

The problem seems to be fixed.
When I thoroughly checked the adapter and switch, the adapter seems to be fixed in switch, but I thought that may be it is not actually fixed and there may be some electrical sparking (my guess). So I changed the adapter as well as switch. And now for two days the problem did not happen
However, it is too early to predict anything. Only time will tell the truth

Now I can safely say that the problem is resolved which it was caused by adapter.

you know it is the case (at least it looks like at this moment) where we could not help you. Unfortunately an adapter can cause such problems, but it cannot be easily handled by a forum like this.
Anyway I have to say I'm glad you could finally find the reason. You made it.

michaelk mentioned failing adapter. so technically i got help from this forum