The fact is, both sides can be right. Take the following example:

School receives $100 donation to the sports program with no restrictions on where the money goes. The school only had 2 sports and elects to spit the money evenly. Therefore, each sport gets $50 into its individual budget / coffers.

No say the same school actually received 2 donations of $50 each. However, on one donation, there is a stipulation that all funds must go to sport A. The second donation has no stipulations.

The school agrees to allocate the $50 donation exclusively to sport A. Consequently, Sport A gets $50 into its budget / coffers. Thus, the stipulation has been complied with.

The school then decides to allocate the other $50 donation with no stipulations 100% to sport B. Therefore, sport B gets $50 into its budget / coffers.

All stipulations have been met; but nothing has really changed and the amount of money the school wanted to go to each sport has not changed. Thus, yes, one can donate money with stipulations and the school will accept. However, do not think that as a result of that donation and stipulation anything has really changed with respect to the funding of your targeted sport. Most likely your stipulation just results in a shuffling of the allocation from the other funds.