Can you have heart failure without a myocardial infarction?

Yes: An example is uncontrolled hypertension (high blood pressure) can cause heart failure without myocardial infarction (heart attack) happening at the same time.

The quick answer is : Yes. Heart failure refers to the inability of the heart to pump properly and thus resulting in accumulation of fluid in the lung. This is called systolic heart failure. There is also diastolic or right sided heart failure and those symptoms are a bit different. Myocardial infarction refers to poor blood supply to the heart resulting in the death of heart tissue. This could result in heart failure.