According to caste-system based on Manu-smriti, only the first three categories i.e., Brahmins, Kshatriyas, Vaishyas are allowed to wears the sacred thread.
Why are Sudras not entitled to wear the sacred thread and learn Vedas?
When a person wears a sacred thread he must follow certain rules & should remain pious all the time. Everyone can't follow all these rules.
In the past era, The people were divided in different Varna according to the kind of the work they did. Since Sudras were the people who did odd jobs. That's why they not used to wear sacred thread.
But now there are many Communities like Gayatri Pariwar , follwers of these communities wear The sacred Thread irrespective of their varna or castes.
This should be very clear that now whoever wants to wear sacred threads may wear it, if he is ready to follow the rituals.