Both effects operate independently.
A good example is the GPS signal. The GPS clocks need to be accurate to within $30ns$ to achieve the required positioning accuracy. However, GPS satellites orbit the earth; in other words they go fast, and at high altitude. Their speeds slows down time, and the reduced gravity makes it flow faster. Hence there are 2 adjustments that need to be made.
- Their speed (almost $14,000km/h$)implies the clocks will be slow by about $7\mu$$s$
- Because of their altitude (about $20,000km$), their clocks will run fast by about $45\mu$$s$ per day.
As a result, the adjustment required is $45-7=38\mu$$s$ per day. This might not sound like much, but remember that light travels at $300,000km/s$, or $11.4km$ in $38\mu$$s$. In other words, without this adjustment, your position would drift more than $300km$ per month!
See here for more information about GPS satellites.