Always be aware of the tense of the verbs in your sentence. My teachers always encouraged (past) me to do the best that I could (past) and that helped (past) me a lot. In this sentence, the verb tenses are consistent, therefore this sentence is correct. My teachers always encouraged (past) me to do the best that I can (present) and that helped (past) me a lot. The verb tenses in this sentence are inconsistent, which is incorrect. If you would like to say that your teachers still do encourage you and this helps you, this would be the correct sentence to use: My teachers encourage me to do my best, which helps me a lot. I hope this helps you! :) Let me know if you need any further explanation.

Can and could have almost the same meaning and are interchangeable a lot of the time. If you want to get technical about it, then here is the difference: CAN: Can is something you are capable of doing right now, when you have what it takes at this very moment to complete the task. Correct usage: "I can help you" "She can sing very well." Incorrect usage: "If I was there then I can do that" COULD: Could is something you are capable of doing, IF you were in a particular situation. While "can" implies immediacy, when using "could" you are not always talking about doing something right now. Correct usage: "I could help you." "I could help you, but I'm not." "If I was there, then I could do that." TRICKY POINT: A lot of times, "can" be replaced by "could", but "could" cannot always be replaced by "can". Often, when you "can" do something right away, you "could" do something if you were given the opportunity. I.e. If you can help your friend right away, then you could help your friend if she needed help. HOWEVER, the word "could" is not always replaceable by the word "can". Just because you "could" do something in a particular environment doesn't mean you "can" do something right away. For example, "If I was given a space suit, I could walk on the moon" is correct, while "If I was given a space suit, I can walk on the moon" is not. YOUR EXAMPLE: "My teachers always encouraged me to do the best that I can and that helped me a lot." This means your teachers encouraged you to do the best, right away, what you are immediately capable of. This is correct english. "My teachers always encouraged me to do the best that I could and that helped me a lot." If your teacher wants you to do what you are immediately capable of, then your teacher might expect you to work when put into a particular situation. This is also correct English. The difference between your two sentences is negligible and they are correct sentences with almost the same meaning.