The Second We Start Losing Multiple Probes at the Same Distance from the Sun the Jig is Up
It would take approximately 30 years of "design/build time" along with the travel time it takes probes to reach the barrier for the world to consider aliens the primary cause of probe loss.
Ash has a lovely list of all the things scientists would think when the first probe is lost. Space dust, random failure of receiver/emitter and so on. However you can rule out multiple problems per launch, and indeed scientists being scientists they probably would. Once you start losing multiple probes at the same range the suspicions are going to pile up pretty quickly. Especially once people start aiming for other parts of the sky and getting the same result.
You sent a probe to Alpha Centauri. The probe loses signal a Light-day out from the sun(122ish AU, roughly the distance from the sun to the edge of the solar system). So you boost the power and harden the systems and send out another probe, had to be something mechanical. If someone's real paranoid you'll also increase the frequency of reporting as you near the 122 AU limit. Second wave probe still goes dark a light-day out. At this point the head scratching increases. But still, can't be aliens, plenty of other things could go wrong! So you do it again, increasing the probe's ability to gather data about their trip, just in case. You lose the probe again.... exactly where the last ones were lost.
Paranoia in the scientific community is now REAL high. You fire off another two probes, of a simpler design, at slightly different patches of the sky because at this point you don't want to waste multi-billion dollar solar-system-scanning probes when what you REALLY want to know is Why Do The Probes Die at 122 AU. You have them continually broadcasting once they get close to The Limit. They go dark at the exact same distance as the last set.
Governments are now Very Concerned, and it wouldn't surprise me at all if the word aliens started getting thrown around. Another pair of probes are launched, this time in tandem, with one following in visual range of the lead probe. The lead probe hits 122 AU, and dissolves. The trailing probe sends back the intel, and is lost soon after. Everyone now must admit that we live in some sort of Galactic Zoo.
So that's 5 iterations from First Loss to cue hand gesture Aliens. You could shrink it to maybe 3 or increase it a few more times, but scientists are real smart and probes are build with every-conceivable-failsafe-and-variable in mind today, so it doesn't seem likely you could keep doing this ad-nauseum.
Where does that leave us? It can be a decade to design, build, and launch a probe, and 5 years seems like a pretty fast turnaround. Plus travel time, which is whatever you make of it. Even if you wanted to launch the same probe again you probably wouldn't, just because of how bureaucracy works. (When probes crash they tend to add on new tech to the next probe even though it's trying to do the exact same thing the lost one was, rather than just re-build the already designed probe. Usually because the new tech is miles better but it still takes redesign time.) So say 5 years between first loss and second launch. Then 10 years between the second and third wave (because at this point you're essentially designing new probes) Then 5 years again (because you're using third-wave design, more or less, just aimed at another spot) then 10 years for the final pair.
Once you start factoring in travel time though it could be a century or more (Space is BIG, takes a long time to get places) before your poor Terrans realize what's what though, and that seems like plenty of time for happy and hopeful colonies on Mars or Europa or wherever. If your travel times are more like a year or two between probes it wouldn't be too hard to imagine budget fights meaning that Starshot II takes 20 years to be approved because hey, the guys wanting to send a probe into Neptune's atmosphere have a REALLY great proposal, or Why Do Probes Die At 122AU Probe I never getting off the ground because Starshot I and II bankrupted the company and why should some OTHER company figure it out when mining the Asteroid Belt is a more lucrative idea? Conversely if you have some competing companies/nations and 3 different Starshot probes are launched at Alpha Centauri a month apart who then get into a Moon-Race-esq contest to figure out WHY they were lost you could come to the "Alien Zoo" conclusion within a decade, minus travel-time.