Is using multiple unique constrains on a single table considered bad design?
There is always Sixth, or Domain Key, Normal Form. Here each non-key column becomes it own table. So 3NF table T(Key, Col1, Col2, ..) becomes T1(Key, Col1), T2(Key, Col2) etc. Those new tables which require uniqueness can have it declared.

I think having multiple unique constraints on a table is perfectly OK, however. Take for example a table of countries. This would have, say, an ID, the name, the ISO code, the capital city, and some others. Each of those first four will be unique. Moreover, if we want our system to rely on each being unique I believe we should define unique constraints on each. This enforces truths about the data on which all consumers can rely.