A bank is always willing to take your money, so, yes, one can add money to a CD account. In fact, it is a good thing to do so since the more money one has in an account, the more one earns from interest. Saving should be something everyone strives to do.
According to the contract for Bank CD's, you can only deposit or withdraw money when your CD is due, which is how long you took it out for. 6 months to 2 years. Otherwise you will pay a penalty. You usually get a letter prior to your due date and have a grace period when to adjust.
In most cases you can not add to a bank cd after opening it. However some bank CD's will allow additions and some will allow additions during the time for renewal. Yet in most cases the CD is locked for the life of the CD with the interest rate with no additions allowed.
A bank certificate of deposit is an agreement between you and the bank that says you will allow them to use your money for a specific period of time in return for their paying you a fixed amount of interest. You may not add or remove funds to the account before the CD expires without
incurring a penalty