Translation:The company believed in me and I cannot fail.
Definitely one example where using the simple past is not appropriate, because it indicates that the belief was in the past and says nothing about current belief. "Has believed" instead states past belief which continues into the present - and THAT is the intent of the sentence. Past simple translation is just wrong here.
The English phrase "I can't fail" can be ambiguous. In this sentence, it most likely means "I must not fail" but in other contexts it could mean "it is not possible for me to fail".
Is there similar ambiguity in Italian, or would you use "non devo fallire" if you meant "I must not fail" (or else I'll suffer some consequence or other)?
Mixing tenses like this remains extremely awkward, even if we translate as "has believed". It could possibly make sense if the sentence contained context like "when ....... occurred, the company (has) believed in me, so now ....... is happening, I cannot fail". Otherwise, the only way to interpret the sentence, IMHO, is that the company has confidence/faith in me and I can't fail.