Does Linux instance have the connection limit?

Short answer: No, there aren't any limits imposed on connections other than what the operating system does.

However. T-series instance work on a CPU credit system. When they aren't busy they build up credits. When they are, they draw down those credits and if the credits run out the performance level returns to a baseline which may not be enough to service your requests in a timely manner.

You can monitor your CPU credits which would be a good thing to look at in this case.

If you are running out of CPU credits three suggestions:

1. Use a larger T-sized instance. This will cost more but it also comes with a higher baseline performance. Not a great suggestion if your performance issue happens with low frequency as you're paying for capacity you don't need.
2. Choose Unlimited mode for your instance. If it runs out of CPU credits we will give you additional CPU credits - for a charge. This is a good option if you only see this issue occasionally but if it is happening more often then the charges may be higher.
3. Choose a non-T series instance that doesn't rely on CPU credits. This has a higher cost but now you don't have to worry about performance changing under load.

This is all assuming that CPU is the problem in this case. It may not be. You should be looking at memory usage, disk I/O and other system metrics to determine why performance suffers when you have many requests.