I'm not a fan of these types of questions or responses because they do not offer universal solutions. You will come across graphs that show a higher bar or cutoff indicating higher sensitivity and lower specificity (opposite of what Phlostein said).
However, Phlostein's explanation is absoluteyl correct, its in how he defined the cutoff that is important. The keyword in Phlostein's explanation is "strict" criteria.
The OP did not indicate what he meant by cutoff. Its possible this is the generally accepted definition but I am not certain.
OP, your best bet is to look at a graphical representation-which FA has an excellent example of, to get a good understanding of what happens when you change the acceptable levels for sensitivity and specificity detection.